Integral geometry in complex space forms

Integral geometry in complex space forms
Judit Abardia Bochaca
October 2009
Memòria presentada per aspirar al grau de
Doctor en Ciències Matemàtiques.
Departament de Matemàtiques de la Universitat
Autònoma de Barcelona.
Directors:
Eduardo Gallego Gómez i Gil Solanes Farrés

CERTIFIQUEM que la present Memòria ha estat realitzada per na
Judit Abardia Bochaca, sota la direcció dels Drs. Eduardo Gallego
Gómez i Gil Solanes Farrés.
Bellaterra, octubre del 2009
Signat: Dr. Eduardo Gallego Gómez i Dr. Gil Solanes Farrés

Contents
Introduction 1
1 Spaces of constant holomorphic curvature 7
1.1 First definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Projective model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.5 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.6 Structure of homogeneous space . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Moving frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Totally geodesic submanifolds . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Geodesic balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Space of complex r-planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.1 Expression for the invariant density in terms of a parametrization . . . 20
1.5.2 Density of complex r-planes containing a fixed complex q-plane . . . . . 25
1.5.3 Density of complex q-planes contained in a fixed complex r-plane . . . . 25
1.5.4 Measure of complex r-planes intersecting a geodesic ball . . . . . . . . . 25
1.5.5 Reproductive property of Quermassintegrale . . . . . . . . . . . . . . . . 26
2 Introduction to valuations 29
2.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Hadwiger Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Alesker Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Valuations on complex space forms . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Smooth valuations on manifolds . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Hermitian intrinsic volumes . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.3 Relation between Hermitian intrinsic volumes and the valuations given
by Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.4 Other curvature integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.5 Relation between the Hermitian intrinsic volumes and the second fundamental
form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Average of the mean curvature integral 45
3.1 Previous lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Integral of the r-th mean curvature integral . . . . . . . . . . . . . . . . . . . . 48
3.3 Mean curvature integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Reproductive valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
v

vi
Contents
3.5 Relation with some valuations defined by Alesker . . . . . . . . . . . . . . . . . 58
3.6 Example: sphere in CK 3 (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Gauss-Bonnet Theorem and Crofton formulas for complex planes 61
4.1 Variation of the Hermitian intrinsic volumes . . . . . . . . . . . . . . . . . . . . 61
4.2 Variation of the measure of complex r-planes intersecting a domain . . . . . . . 66
4.3 Measure of complex r-planes meeting a regular domain . . . . . . . . . . . . . . 68
4.3.1 In the standard Hermitian space . . . . . . . . . . . . . . . . . . . . . . 68
4.3.2 In complex space forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Gauss-Bonnet formula in CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Another method to compute the measure of complex lines . . . . . . . . . . . . 76
4.5.1 Measure of complex lines meeting a regular domain in C n . . . . . . . . 76
4.5.2 Measure of complex lines meeting a regular domain in CP n and CH n . . 77
4.6 Total Gauss curvature integral C n . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Other Crofton formulas 81
5.1 Space of (k, p)-planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1.1 Bisectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Variation of the measure of planes meeting a regular domain . . . . . . . . . . 84
5.3 Measure of real geodesics in CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4 Measure of real hyperplanes in C n . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 Measure of coisotropic planes in C n . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.6 Measure of Lagrangian planes in CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . 90
Appendix 93
Proof of Theorem 4.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Bibliography 103
Notation 107
Index 109

Introduction
Classically, integral geometry in Euclidean space deals with two basic questions: the expression
of the measure of planes meeting a convex domain, the so-called Crofton formulas; and the
study of the measure of movements taking one convex domain over another fixed convex
domain, the so-called kinematic formula.
In the Euclidean space R n , we denote by L r a totally geodesic submanifold of dimension
r, and we call it r-plane. We denote the space of r-planes by L r . This space has a unique
(up to a constant factor) density invariant under the isometry group of R n , denoted by dL r .
Then, given a convex domain Ω ⊂ R n with smooth boundary, the expression of the measure
of r-planes meeting a convex domain is given by
∫
L r
χ(Ω ∩ L r )dL r = c n,r M r−1 (∂Ω), (1)
where c n,r only depends on the dimensions n, r, and M r−1 (∂Ω) denotes the integral over ∂Ω
of the (r − 1)-th mean curvature integral.
Thus, the mean curvature integrals appear naturally in the Crofton formula. A classical
known property of mean curvature integrals is the following
∫
M (r)
i
(∂Ω ∩ L r )dL r = c ′ n,r,iM i (∂Ω) (2)
L r
where c ′ (r)
n,r,i only depends on the dimensions n, r, i, and M
i
(∂Ω ∩ L r ) denotes the i-th mean
curvature integral of ∂Ω ∩ L r as a hypersurface in L r
∼ = R r . From (2), it is said that mean
curvature integrals satisfy a reproductive property.
On the other hand, the kinematic formula in R n is expressed as follows. Let Ω 1 and Ω 2 be
two convex domains with smooth boundary, let O(n) := O(n) ⋉ R n denote the isometry group
of R n , and let dg be an invariant density of O(n). Then,
∫
O(n)
χ(Ω 1 ∩ gΩ 2 )dg =
n∑
c n,i M i (∂Ω 1 )M n−i (∂Ω 2 ). (3)
i=0
The previous three formulas were extended to projective and hyperbolic spaces (cf. [San04]),
i.e. they are known in the spaces of constant sectional curvature k. The generalization of integral
(2) does not depend on k but in the expression (1) for projective and hyperbolic space
appear other terms, depending on k. Moreover, its expression depends on the parity of the
dimension of the planes. If r is even, then
∫
L r
χ(Ω ∩ L r )dL r = c n,r−1 M r−1 (∂Ω) + c n,r−3 M r−3 (∂Ω) + · · · + c n,1 M 1 (∂Ω) + c n vol(Ω), (4)
and if r is odd
∫
L r
χ(Ω ∩ L r )dL r = c n,r−1 M r−1 (∂Ω) + c n,r−3 M r−3 (∂Ω) + · · · + c n,2 M 2 (∂Ω) + c n vol(∂Ω), (5)
1

2 Introduction
where c n,j depends on the dimensions n and j and are multiples of k n−j .
The facts that the expression depends on the parity, and that we study an integral of
the Euler characteristic, remain us the Gauss-Bonnet formula in spaces of constant sectional
curvature, which also depends on the parity of the ambient space. We recall here this formula
in a space of constant sectional curvature k and dimension n.
If n is even, then
M n−1 (∂Ω) + c n−3 M n−3 (∂Ω) + · · · + c 1 M 1 (∂Ω) + k n/2 vol(Ω) = vol(S n−1 )χ(Ω),
and if n is odd,
M n−1 (∂Ω) + c n−3 M n−3 (∂Ω) + · · · + c 2 M 2 (∂Ω) + k (n−1)/2 vol(∂Ω) = vol(Sn−1 )
χ(Ω)
2
where c i depends only on the dimensions n, i and are multiples of the sectional curvature k.
Now, using the expression (2) and the Gauss-Bonnet formula, we get (4) and (5).
The goal of this work is generalize formulas (1) and (2) in the standard Hermitian space
C n , in the complex projective space and in the complex hyperbolic space, denoted by CK n (ɛ)
with 4ɛ the holomorphic curvature of the manifold (see Section 1.1).
In order to achieve this goal, we use the notion of valuation in a vector space V , a realvalued
functional φ from the space of convex compact domains K(V ) in V to R satisfying the
following additive property
φ(A ∪ B) = φ(A) + φ(B) − φ(A ∩ B)
whenever A, B, A ∪ B ∈ K(V ).
The first examples of valuations are the volume of the convex domain, the area of the
boundary, and the Euler characteristic. Other classical examples of valuations are the socalled
intrinsic volumes. They are defined from the Steiner formula: given a convex domain
Ω ⊂ R n , if we denote by Ω r the parallel domain at a distance r, the Steiner formula relates
the volume of Ω r with the so-called intrinsic volumes V i (Ω) by
vol(Ω r ) =
n∑
r n−i ω n−i V i (Ω)
i=0
where ω n−i denotes the volume of the (n − i)-dimensional Euclidean ball with radius 1 (cf.
Proposition 2.1.3).
If Ω ⊂ R n is a convex domain with smooth boundary, then intrinsic volumes satisfy
V i (Ω) = cM n−i−1 (∂Ω),
and they are the natural generalization of the mean curvature integrals for non-smooth convex
domains.
Hadwiger in [Had57] proved that all continuous valuations in R n invariant under the isometry
group of R n are linear combination of the volume of the convex domain, the area of
the boundary, and the intrinsic volumes (see Section 2.2.1). This result has as immediate
consequence formulas (1), (2) and (3).
Alesker in [Ale03] proved that the dimension of the space of continuous valuations in C n
invariant under the holomorphic isometry group of C n is ( )
n+2
2 and gave a basis of this space.
In the recent paper of Bernig and Fu, [BF08], there are given other basis of valuations in C n .
In particular, the Hermitian intrinsic volumes are defined (see Section 2.4.2). These are the
valuations we will use to work with. The fact that the dimension of the space of continuous
valuations invariant under the isometry group of C n is bigger than the one of R 2n is not

3
surprising if we recall that the holomorphic isometry group of C n , U(n), is smaller than the
isometry group of R 2n , O(2n).
Hermitian intrinsic volumes are a kind of generalization of mean curvature integrals, but
taking into account that C n has a complex structure which defines a canonical vector field on
hypersurfaces. Indeed, at each point x of a hypersurface, if we consider the normal vector, and
we apply the complex structure, then we get a distinguished vector JN in the tangent space
of the hypersurface at x. Moreover, the orthogonal space to JN in the tangent space defines
a complex space of maximum dimension, n − 1.
So, if S is a smooth hypersurface in C n , we can consider the integral
∫
k n (JN)dx
S
where k n (JN) denotes the normal curvature in the direction JN, and this is a valuation in
C n . Other valuations related to normal curvature of the direction JN appear as elements in
of Hermitian intrinsic volumes basis.
The notion of valuation can be also defined in a differentiable manifold (see Definition
2.4.1). In real space forms the volume of a convex domain and the area of its boundary are
valuations. But, it is not known an analogous result to Hadwiger Theorem in these spaces.
The definition of the Hermitian intrinsic volumes can be extended to other space of constant
holomorphic curvature. We denote by {µ k,q } the Hermitian intrinsic volumes. The subscript
k denotes the degree of the valuation (see Section 2.4.2).
In order to give a similar expression of (1) and (2) in the spaces of constant holomorphic
curvature, we need to describe the integration space. Note that in spaces of constant sectional
curvature we integrate over the space of r-planes, i.e. totally geodesic submanifold of fixed dimension.
In spaces of constant holomorphic curvature, complete totally geodesic submanifolds
are classified. If ɛ ≠ 0 they are complex submanifolds isometric to CK r (ɛ) ⊂ CK n (ɛ), with
1 ≤ r < n or totally real submanifolds isometric to RK q (ɛ) ⊂ CK n (ɛ) with 1 ≤ q ≤ n, where
RK q (ɛ) denotes the space of constant sectional curvature ɛ. For ɛ = 0 there are other totally
geodesic submanifolds. We denote the space of complex planes with complex dimension r,
1 ≤ r < n, by L C r , and the space of totally real planes of maximum dimension n by L R n, the
so-called Lagrangian manifolds.
In this work, we obtain a Crofton formula for complex r-planes and Lagrangian planes
∫
( ) n − 1
χ(Ω ∩ L r )dL r = vol(G C n−1,r)
−1· (6)
L C r
· (
n−1
∑
k=n−r
ɛ k−(n−r) ω 2n−2k
( n
k
+ ɛ r (r + 1)vol(Ω)),
) −1
⎛
⎝
r
∑k−1
q=max{0,2k−n}
( 2k−2q
)
⎞
k−q
4 k−q µ 2k,q (Ω) + (k + r − n + 1)µ 2k,k (Ω) ⎠
and
∫
L R n
⎛
· ⎝
∫
L R n
χ(Ω ∩ L)dL = vol(G 2n,n)ω n
n!
n−1
2∑
q=0
( ) 2q − 1 −1
4 q−n
q − 1 2q + 1 µ n,q(Ω) if n is odd, (7)
χ(Ω ∩ L)dL = vol(G 2n,n)
· (8)
n!
n
2∑
( ) 2q − 1 −1
4 q−n n
⎞
ω n
2∑
( ) n −1
q − 1 2q + 1 µ n,q(Ω) + ɛ i 2 −n+1 ω n−2i
n
2 + i µ
n + 1 n+2i,
n
2 +i (Ω) ⎠ if n is even,
q=0
i=1

4 Introduction
where ω i denotes the volume of the i-dimensional Euclidean unit ball.
Previous formulas have more addends that the corresponding ones in the spaces of constant
sectional curvature, but they are similar. If ɛ = 0, i.e. in C n also appear all the valuations
with the corresponding degree. If ɛ ≠ 0 the notion of degree of a valuation has no sense but
there is a similitude with the expression in spaces of constant sectional curvature comparing
the subscripts of the valuations.
In order to get these expressions we use a variational method. That is, we take a smooth
vector field X defined on the manifold and we consider its flow φ t . We prove the following
formula of first variation
∫
∫ ∫
d
dt∣ χ(φ t (Ω) ∩ L r )dL r = 〈X, N〉
σ 2r (II| V )dV dp
t=0 G C n−1,r (Dp)
L C r
∂Ω
where N is the exterior normal field, D is the distribution in the tangent space at ∂Ω orthogonal
to JN, and σ 2r (II| V ) denotes the 2r-th symmetric elementary function of the second
fundamental form II restricted to V ∈ G C n−1,r (D p), the Grassmannian of complex planes with
complex dimension r inside D p .
On the other hand, we get an expression of the variation of valuations µ k,q using the method
in [BF08].
Comparing both variations and solving a system of linear equations we obtain the result.
Using the same variational method we also obtain a Gauss-Bonnet formula for the spaces
of constant holomorphic curvature. It is known that the variation of the Euler characteristic
is zero. Thus, we can express it as a sum of Hermitian intrinsic volumes such that its variation
vanishes. The obtained Gauss-Bonnet formula is the following
⎛
n−1
∑
ω 2n χ(Ω) = (n + 1)ɛ n ɛ c ω 2n−2c (n − c)
vol(Ω) +
n ( ) ⎝
n−1
c
c=0
∑c−1
q=max{0,2c−n}
( 2c−2q
)
⎞
c−q
4 c−q µ 2c,q + (c + 1)µ 2c,c
⎠.
In spaces of constant sectional curvature k, Solanes in [Sol06] related the measure of planes
meeting a domain with the Euler characteristic of the domain
In CK n (ɛ), we get
ω n χ(Ω) = 1 n M n−1(∂Ω) +
2k ∫
nω n−1
L n−2
χ(Ω ∩ L n−2 )dL n−2 .
ω 2n χ(Ω) = 1
∫
2n M 2n−1(∂Ω) + ɛ χ(Ω ∩ L n−1 )dL n−1 +
L C n−1
n∑
j=1
ɛ j ω 2n
µ 2j,j (Ω).
ω 2j
(9)
The analogous expression to (2), it is given when we integrate the mean curvature integral
over complex r-planes. The obtained expression for a compact oriented (possible with
boundary) hypersurface S of class C 2 is
∫
L C r
M (r)
1 (S∩L r)dL r = ω 2n−2vol(G C n−2,r−1 ) ( n −1 (
(2n − 1)
2r(2r − 1) r) 2nr − n − r ∫
M 1 (S) +
n − r
where k n (JN) denotes the normal curvature in the direction JN ∈ T S.
S
)
k n (JN)
(10)

5
To get this result, first we obtain, using moving frames, the following intermediate expression
(for any mean curvature integral). If r, i ∈ N such that 1 ≤ r ≤ n and 0 ≤ i ≤ 2r − 1,
then
∫
M (r)
i
(S ∩ L r )dL r (11)
L C r
( ) 2r − 1 −1 ∫
=
i
S
∫RP 2n−2 ∫
G C n−2,r−1
|〈JN, e r 〉| 2r−i
(1 − 〈JN, e r 〉 2 ) r−1 σ i(p; e r ⊕ V )dV de r dp,
where e r ∈ T p S unit vector, V denotes a complex (r − 1)-plane containing p and contained in
{N, JN, e r , Je r } ⊥ , σ i (p; e r ⊕ V ) denotes the i-th symmetric elementary function of the second
fundamental form of S restricted to the real subspace e r ⊕ V and the integration over RP 2n−2
denotes the projective space of the unit tangent space of the hypersurface.
In order to complete the generalization of equation (1) in CK n (ɛ), it remains to study the
measure of (non-maximal) totally real planes. These are the other totally geodesic submanifolds
of CK n (ɛ), ɛ ≠ 0. Using the same techniques as in the rest of this work, it does not
seem possible to solve this case since we cannot obtain enough information of the variational
properties of the measures of totally real planes meeting a domain in CK n (ɛ).
On the other hand, it would be interesting to extend formula (10) to i ∈ {2, . . . , 2r − 1}.
Next we explain the organization of the text.
Chapter 1 contains a description of the spaces of constant holomorphic curvature. We
review its definition and describe some of the most important submanifolds, i.e. the totally
geodesic submanifolds, the geodesic spheres, and the complex planes. In this chapter we also
recall the method of moving frames, which will be used along this text. Using moving frames,
we give an expression for the density of the space of complex planes. Finally, we prove that
integral ∫ L χ(Ω ∩ L r)dL C r satisfies a reproductive property.
r
Chapter 2 is devoted to the study of valuations in C n and in the spaces of constant holomorphic
curvature. First of all, we review the concept and the main properties of the valuations
on R n together with the Hadwiger Theorem (which characterize all continuous valuations in
R n invariants under the isometry group). An analogous Hadwiger Theorem in C n is stated.
Finally, we define the used valuations in this work in spaces of constant holomorphic curvature,
and we give new properties and relations with other valuations also important in the
next chapters.
Chapter 3 gives a proof of (10). First of all, we prove some geometric lemmas and we
obtain the expression for the mean curvature integrals over the space of complex planes in
terms of an integral over the boundary of the domain given in (11). This expression will be
fundamental to attain the goal of this chapter. As a corollary of (10) we characterize the
valuations of degree 2n − 2 satisfying a reproductive property in C n , and we give the relation
among different valuations defined by Alesker (already reviewed in Chapter 2). The results of
this chapter are contained in [Aba].
In Chapter 4 we obtain the measure of complex planes intersecting a domain in the spaces of
constant holomorphic curvature in terms of the Hermitian intrinsic volumes defined at Chapter
2. We also give an expression of the Gauss-Bonnet formula in terms of these valuations. In
order to get these expressions we use a variational method. First, we obtain an expression for
the variation of the measure of complex planes and for the Hermitian intrinsic volumes. In
this chapter, we verify the certainty of (6). A constructive proof, where we find the constants
in the expression is given in the appendix. As a corollary, we express the total mean Gauss
curvature in C n also in terms of the Hermitic intrinsic volumes. Finally, we relate Chapters 3

6 Introduction
and 4 obtaining another method to compute the measure of complex lines meeting a domain.
The results of this chapter are contained in [AGS09].
Chapter 5 studies the measure of another type of planes meeting a domain in C n , the socalled
coisotropic planes. These planes are the orthogonal direct sum of a complex subspace
of complex dimension n − p and a totally real subspace of dimension p. Totally real planes of
maximum dimension and real hyperplanes are particular cases of this type of planes. Using
similar techniques as in Chapter 4 we give an expression for the measure of planes of this type
meeting a domain. For the spaces of constant holomorphic curvature we prove (7) and (8),
which give the measure of totally real planes of maximum dimension, the so-called Lagrangian
planes.
The appendix contains the constructive proof of (6) and (9). That is, we give the method
that allowed us to obtain the constants appearing in these expressions. This proof consists,
at a final instance, to solve a linear system obtained from the study of the variation of both
sides of the expressions, as it is detailed in Chapter 4.
Acknowledgments
In these final lines, I would like to express my gratitude to all people who, in some way, allowed
me to convert this work into a Thesis. Specially, I would like to mention my first director,
Eduardo Gallego, my codirector, Gil Solanes, for their patient and support; and the Group
of Geometry and Topology of the Universitat Autònoma de Barcelona, in particular Agustí
Reventós, to introduce me to the “geometry world”. I would also like to thank Andreas Bernig
for all his advices, and the members of the Department of Mathematics in the University of
Fribourg to offer me such nice work ambience during my stay. Although I am not going to
write down a list with the name of all mathematicians who contributed with their enlightening
conversations to the completion of this work, and hopefully, to some future work, I would be
very satisfied if they feel identified in these lines.
Outside from the direct relation to this work, I would like to thank my colleagues of
doctorate, specially my colleagues in the room, and also in the geometry and topology area,
in Barcelona, but also in Fribourg. In the same way I would like to thank all the persons I
have met during these years and become my friends.
Finally, I thank my parent’s support, not only in these years of doctorate.
This work has been written under the support of the “Departament d’Universitats, Recerca
i Societat de la Informació de la Generalitat de Catalunya”, the European Social Found and
the Swiss National Found.

Chapter 1
Spaces of constant holomorphic
curvature
1.1 First definitions
In this section we introduce the spaces of constant holomorphic curvature, also called complex
space forms, and give the properties we shall use along this work. First of all, we recall some
basic definitions.
Definition 1.1.1. Let M be a differentiable manifold. M is a complex manifold if it has an
atlas such that the change of coordinates are holomorphic, that is, {(U α , φ α )} is an atlas with
{U α } an open covering of M and φ α : U α → C n homeomorphisms such that φ β ◦ φ −1
α are
holomorphic in its domain of definition.
Examples
(i) The vector space C n is a complex manifold of complex dimension n.
(ii) The complex projective space CP n is a complex manifold of complex dimension n.
The complex projective space can be defined analogously to the real projective space.
Let us consider in C n+1 \{0} the equivalence relation which identifies the points differing
by a complex multiple. Then, we take as an atlas the open sets {U 0 , ..., U n } such that
U j = {(z 0 , ..., z n ) ∈ C n+1 | z j ≠ 0}
and for every U j we take the map φ j (z 0 , ..., z n ) = (z 0 /z j , ..., z j−1 /z j , z j+1 /z j , ..., z n /z j )
which is a homeomorphism. It can be proved that the change of coordinates are holomorphics.
Definition 1.1.2. Let M be a complex manifold. A linear map J : T x M → T x M is an almost
complex structure of M if for each x ∈ M, the restriction of J at T x M satisfies J x : T x M →
T x M, J 2 = −Id, and J varies differentially on M.
Note that any complex manifold admits an almost complex structure. Indeed, the tangent
space of a complex manifold has a complex vector space structure, so the map “multiply by i”
is well-defined and satisfies that applied twice is the map −Id. We call this canonical almost
complex structure complex structure.
Definition 1.1.3. Let V be a complex vector space and let u, v ∈ V . It is said that h :
V × V → C is an Hermitian product on V if
7

8 Spaces of constant holomorphic curvature
1. it is C-linear with respect to the first component,
2. h(u, v) = h(u, v).
Remark 1.1.4. From the properties of a Hermitian product, it follows that if λ ∈ C then
h(u, v) = λh(u, v). Indeed, by definition we get the following equalities
h(u, λv) = h(λv, u) = λh(v, u) = λh(u, v).
Definition 1.1.5. Let M be a differentiable manifold with complex structure J and a Riemannian
metric g. Then, g is called a Hermitian metric if it is compatible with the complex
structure, i.e. it satisfies g x (Ju, Jv) = g x (u, v) for every x ∈ M and u, v ∈ T x M.
Definition 1.1.6. Let M be a complex manifold with complex structure J and Hermitian
metric g. The 2-form ω defined by
is the Kähler form.
ω(u, v) = g(u, Jv), ∀ u, v ∈ T x M
Remark 1.1.7. Given a complex manifold M with complex structure J and a Hermitian product
defined on T M, we get a Hermitian metric on M from the real part of the Hermitian product,
and a Kähler form on M from the imaginary part of the Hermitian product.
Definition 1.1.8. A complex manifold M is called a Kähler manifold if it has a Hermitian
metric such that the Kähler form associated to this metric is closed.
Proposition 1.1.9 ([O’N83] page 326). Let M be a Kähler manifold with connection ∇.
Then,
∇JX = J∇X, ∀X ∈ X(M).
Definition 1.1.10. A subspace W ⊂ T x M of complex dimension 1 is a complex direction or
a holomorphic section of the tangent space if W is invariant under J, i.e. JW = W .
If w ≠ 0 ∈ W , then the vectors {w, Jw} constitute a basis of W , as a real subspace.
Definition 1.1.11. The holomorphic curvature is the sectional curvature of holomorphic sections.
Definition 1.1.12. A space of constant holomorphic curvature 4ɛ of dimension n is a complete,
simply connected Kähler manifold of complex dimension n, such that the holomorphic
curvature is constant and equal to 4ɛ for every point and every complex direction.
Theorem 1.1.13 ([KN69] Theorem 7.9 page 170). Two complete, simply connected Kähler
manifolds with constant holomorphic curvature equal to 4ɛ are holomorphically isometric.
Definition 1.1.14. We denote by CK n (ɛ) any space of constant holomorphic curvature of
dimension n. If ɛ > 0, then it corresponds to the complex projective space CP n , if ɛ < 0, to
the complex hyperbolic space CH n , and if ɛ = 0, to the Hermitian standard space C n .
Spaces of constant holomorphic curvature are also called complex space forms.
Remark 1.1.15. Complex space forms are, in some sense, a generalization of real space forms
(spaces of constant sectional curvature), i.e. the complete simply connected Riemannian manifolds
with constant sectional curvature. Real space forms are (up to isometry) the Euclidean
space R n , the real projective space RP n and the real hyperbolic space H n . The results in this
work extend some of the classical results in integral geometry from real space forms to complex
space forms. Santaló [San52] and Griffiths [Gri78], among others, obtained some results
of classical integral geometry in the standard Hermitian space and in the complex projective
space taking complex submanifolds. In this work, we deal with non-empty domains and, thus,
with real hypersurfaces.

1.2 Projective model 9
Definition 1.1.16. Given two planes Π and Π ′ of real dimension 2 in a vector space with a
scalar product, the angle between the two planes is defined as the infimum among the angles
between a pair of vectors, one in Π and the other one in Π ′ .
Definition 1.1.17. Let Π be a plane with real dimension 2 in the tangent space of a point in a
Kähler manifold with complex structure J. The holomorphic angle µ(Π) is the angle between
Π and J(Π).
Proposition 1.1.18 ([KN69] page 167). Let M be a Kähler manifold with complex structure
J and Hermitian metric g. The holomorphic angle of a plane Π ⊂ T x M, x ∈ M, is given by
where u, v form an orthonormal basis of Π.
cos µ(Π) = |g(u, Jv)|
Remark 1.1.19. The holomorphic angle of a plane takes values between 0 and π/2. In the
extreme cases we have holomorphic planes, when the holomorphic angle is 0; and totally real
planes defined as the planes with holomorphic angle π/2.
In a complex space form, the sectional curvature of any plane can be computed from the
holomorphic curvature and the holomorphic angle of the plane.
Proposition 1.1.20 ([KN69] page 167). Let M be a Kähler manifold with constant holomorphic
curvature 4ɛ. Then, the sectional curvature of any plane Π ⊂ T x M, x ∈ M is given
by
where µ(Π) is the holomorphic angle of the plane Π.
K(Π) = ɛ ( 1 + 3 cos 2 µ(Π) ) (1.1)
Corollary 1.1.21. Sectional curvature of any plane in the tangent space of a point in a
complex space form with constant holomorphic curvature 4ɛ lies in the interval [ɛ, 4ɛ], if ɛ > 0
and in the interval [4ɛ, ɛ], if ɛ < 0.
1.2 Projective model
Along this work, we shall use the projective model of CK n (ɛ), which we describe here briefly
(cf. [Gol99]).
If ɛ = 0, we are considering the standard Hermitian space C n with the standard Hermitian
product. Along this section we suppose ɛ ≠ 0, unless otherwise stated.
1.2.1 Points
Endow C n+1 with the Hermitian product
Define
(z, w) = sign(ɛ)z 0 w 0 +
n∑
z j w j . (1.2)
j=1
H := {z ∈ C n+1 | (z, z) = ɛ}.
H is a real hypersurface of C n+1 (i.e. it has real dimension 2n + 1). We define the points of
CK n (ɛ) as
CK n (ɛ) := π(H)
where
π : C n+1 \{0} → C n+1 \{0}/C ∗ = CP n . (1.3)

10 Spaces of constant holomorphic curvature
Remarks 1.2.1. (i) The fiber of Π for the points in π(H) =: CK n (ɛ) is S 1 . Indeed, let z,
w ∈ H such that π(z) = π(w). By definition of π, we have w = αz. On the other hand,
it holds (z, z) = (αz, αz) = ɛ. Thus, α = e iθ with θ ∈ R and π −1 ([z]) ∼ = S 1 .
(ii) If ɛ > 0, then CK n (ɛ) coincides as a subset with CP n . But, if ɛ < 0, then CK n (ɛ) is an
open set of CP n .
The differentiable structure and the structure of a complex manifold we take in CK n (ɛ) is
the same as the one of an open set of CP n .
1.2.2 Tangent space
The tangent space of a point z ∈ H is
T z H = {w ∈ C n+1 | Re(z, w) = 0}.
The elements in the tangent space of π(z) ∈ CK n (ɛ) are obtained from the image of the
elements in the tangent space of the point z under dπ. Moreover, the kernel of dπ has dimension
1.
The direction that its image under dπ is the null vector is Jz, since it is the tangent
direction to the fiber. (Note that Jz ∈ T z H since Re(z, Jz) = 0.) Indeed, the fiber of a point
[z] is {e iθ z | θ ∈ R}, then
∂(e iθ z)
∂θ ∣ = iz = Jz.
θ=0
Thus, the tangent space at z ∈ H can be decomposed as
T z H = 〈Jz〉 ⊕ 〈Jz〉 ⊥ .
The tangent space at points in CK n (ɛ) coincides with the image by the differential map of
the projection of vectors 〈Jz〉 ⊥ at T z H.
Given a vector v ∈ T π(z) CK n (ɛ) there are infinitely many vectors of T z H such that under
the differential map dπ give the same vector v, but we can distinguish the one lying in 〈Jz〉 ⊥ ,
which is called horizontal lift and we denote by v L . All other vectors are obtained as linear
combination of this vector and a multiple of Jz.
1.2.3 Metric
Let v, w ∈ T π(z) CH n . The Hermitian product at CK n (ɛ) between v, w is defined by
(v, w) ɛ := (v L , w L ), (1.4)
that is, the Hermitian product defined at C n+1 applied to the horizontal lift of the vectors.
The real part of this product gives a Hermitian metric on CK n (ɛ)
〈v, w〉 ɛ := Re(v L , w L ).
This metric coincides with the so-called Fubini-Study metric, if ɛ > 0 and with the so-called
Bergmann metric, if ɛ < 0 (cf. [Gol99, page 74]).
Along this work we denote the Hermitian metric of CK n (ɛ) by 〈 , 〉 instead of 〈 , 〉 ɛ .
Notation 1.2.2. In order to unify the study of the complex space forms, we define, in the
same way as it is classically done in real space forms, the following trigonometric generalized
functions
⎧
sin(α √ ɛ)
√ , if ɛ > 0
⎪⎨ ɛ
sin ɛ (α) = α, if ɛ = 0
sinh(α √ −ɛ)
⎪⎩ √ , if ɛ < 0
−ɛ

1.2 Projective model 11
and
⎧
⎨ cos(α √ ɛ), if ɛ > 0
cos ɛ (α) = 1, if ɛ = 0
⎩
cosh(α √ |ɛ|), if ɛ < 0
cot ɛ (α) = cos ɛ(α)
sin ɛ (α) .
1.2.4 Geodesics
Geodesics in the projective model of CK n (ɛ) are given by the projection of the intersection
points between H and a plane in C n+1 such that it is spanned by a vector corresponding to a
representative in H ⊂ C n+1 of a point z in the geodesic at CK n (ɛ), and a vector u tangent to
the geodesic at z.
Then, the expression of a geodesic at CK n (ɛ) is given by [γ(t)] = [cos ɛ (t)z + sin ɛ (t)u] where
u ∈ 〈Jz〉 ⊥ ⊂ T z H.
The distance between two points in the complex projective and hyperbolic space can be
expressed in terms of the Hermitian product defined at C n+1 .
Proposition 1.2.3 ([Gol99] page 76). Let x, y ∈ CK n (ɛ), ɛ ≠ 0, and let d be the distance
between the two given points. If x ′ and y ′ are representatives of x and y, respectively, in the
projective model, then the distance between the two points is given by
(cos ɛ d(x, y)) 2 = (x′ , y ′ )(y ′ , x ′ )
(x ′ , x ′ )(y ′ , y ′ )
where ( , ) denotes the Hermitian product in C n+1 defined at (1.2).
1.2.5 Isometries
Let us recall the definition of the matrix Lie group U(p, q).
Definition 1.2.4. Let (x, y) = − ∑ p−1
j=0 x jy j + ∑ n
j=p x jy j be a Hermitian product in C n and
p, q ∈ N ∪ {0} such that p + q = n + 1. Then it is defined
U(p, q) = {A ∈ M n×n (C) | (Av, Aw) = (v, w) with v, w ∈ C n }.
The matrix group U(n) coincides with U(0, n), that is, we consider the standard Hermitian
product on C n .
The matrices of P U(n + 1) = U(n + 1)/(multiplication by scalars), if ɛ > 0 (resp. the
matrices of P U(1, n) = U(1, n)/(multiplication by complex scalars), if ɛ < 0) act naturally on
CK n (ɛ). Moreover, they preserve the metric defined in the model since preserve the Hermitian
product defined at C n+1 . Then, the matrices in P U(n + 1) (resp. P U(1, n)) are isometries of
CP n (resp. CH n ).
Proposition 1.2.5 ([Gol99] page 68).
map in C n+1 .
• Every isometry of CK n (ɛ) comes from a linear
• The isometry group of CK n (ɛ) is P U ɛ (n) with
⎧
⎨ C n ⋊ U(n), if ɛ = 0,
U ɛ (n) = U(n + 1) = U(0, n + 1), if ɛ > 0,
⎩
U(1, n), if ɛ < 0.
(1.5)

12 Spaces of constant holomorphic curvature
In order to unify the study of the complex space forms, independently of ɛ, we represent
the elements in the group C n ⋊ U(n) as matrices
( )
1 0
C n . (1.6)
U(n)
The transitivity of the isometry group at different levels is given in the following proposition.
Proposition 1.2.6 ([Gol99] page 70). The isometry group of CK n (ɛ) acts transitively
• on the points in CK n (ɛ),
• on the unit tangent bundle. That is, given (p, v), (q, w) in the unit tangent bundle there
exists an isometry σ such that σ(p) = q and dσ(v) = w.
• on the holomorphic sections (see Definition 1.1.10).
In the following lemma, we give a basis of left-invariant forms of U(p, q). We will prove
that these forms are also right-invariants.
Lemma 1.2.7. Let A = (a 0 , . . . , a m ) ∈ U(p, q), with p, q, m ∈ N∪{0} such that p+q = m+1.
A basis of left-invariant forms in U(p, q) is given by {Re(ϕ jk ), Im(ϕ jk ), Re(ϕ jj )}, 0 ≤ j ≤ k ≤
m, j ≠ k where ϕ ij = (da i , a j ) and (x, y) = − ∑ p−1
j=0 x jy j + ∑ m
j=p x jy j in C m+1 .
Proof. From Definition 1.2.4 of U(p, q) it follows that A ∈ U(p, q) if and only if A −1 = εA t ε
where
( )
−Idp 0
ε =
. (1.7)
0 Id q
In order to find a basis of left-invariant forms we compute A −1 dA with A ∈ U(p, q). If we
denote A = (a 0 , . . . , a m ), then
⎛
⎞
(da 0 , −a 0 ) . . . (da m , −a 0 )
A −1 dA = εA T .
.
εdA =
⎜ (da 0 , −a p−1 ) . . . (da m , −a p−1 )
= (ϕ ij ) ij . (1.8)
⎟
⎝ (da 0 , a p ) . . . (da m , a p ) ⎠
(da 0 , a m ) . . . (da m , a m )
Each entry of this matrix is a 1-form given by ϕ ij = ±(da i , a j ).
Note that each a j is a m-tuple of complex numbers, so that the 1-forms ϕ ij are complexvalued.
In order to find a basis of left-invariant real-valued 1-forms from the entries of the former
matrix we use the following
• (a j , a j ) = ±1 and differentiating
0 = (a j , da j ) + (da j , a j ) = (da j , a j ) + (da j , a j ) = 2Re(da j , a j ).
Thus, ϕ jj = −ϕ jj and each ϕ jj takes only imaginary values.
• (a j , a k ) = 0 if k ≠ j and differentiating
0 = (da j , a k ) + (a j , da k ).
Thus, {
ϕjk = ϕ kj if j ∈ {0, . . . , p − 1} or k ∈ {0, . . . , p − 1}
ϕ jk = −ϕ kj
otherwise.

1.2 Projective model 13
On the other hand, it follows directly from the definition of U(p, q) that its dimension is
(p + q) 2 . Then, {Re(ϕ jk ), Im(ϕ jk ), Re(ϕ jj )}, 0 ≤ j ≤ k ≤ m, j ≠ k constitutes a basis of
left-invariant 1-forms since they generate all the space and there are (p + q) 2 forms.
For ɛ = 0, if we denote the elements A ∈ C n ⋊ U(n) as the matrices of (1.8), we define the
forms
ϕ ij = (da i , a j )
with ( , ) the standard product in C n+1 .
Definition 1.2.8. A group G is said to be unimodular if there exists a volume element of G
left and right-invariant.
Lemma 1.2.9. U(p + q) is a unimodular group.
Proof. From Lemma 1.2.7 we have a basis of left-invariant forms of U(p, q). We prove that
each of these forms is also right-invariant, i.e. it satisfies
R ∗ Bϕ ij (A; v) = ϕ ij (A; v)
∀A, B ∈ U(p, q), v ∈ T A U(p, q).
We use the expression ϕ ij = ±(da i , a j ) and we denote by a k (A) the map taking the k-th
column of a matrix A. Then, if A, B ∈ U(p, q) and v ∈ T A U(p, q)
RBϕ ∗ ij (A; v) = ϕ ij (R B (A); d(R B )(v)) = ±(da i (d(R B )(v)), a j (R B (A))) = ±(da i (vB), a j (AB))
( m∑ ∑p−1
m
= ± − vi k b r k al j br l +
∑ m∑
)
vi k b r k al j br l
(
= ± −
(
= ±
k,l=0 r=0
p−1
∑
k,l=0
δ kl v k i a l j +
m ∑
k,l=p
k,l=0 r=p
δklv k i a l j
∑p−1
m
− vi k a k j + ∑
)
vi k a k j
= ±(da i (v), a j (A)) = ϕ ij (A; v).
k=0
k=p
Then, U(p, q) is a unimodular group since the volume element obtained from the product of the
forms ϕ ij is left-invariant and right-invariant (it is a product of forms with this property).
)
1.2.6 Structure of homogeneous space
Definition 1.2.10. Let (M, g) be a Riemannian manifold. If given any two points x, y ∈ M
there exists an isometry σ of M such that σ(x) = y, then M is a homogeneous space. That is,
a Riemannian manifold is homogeneous if it is a homogeneous space of its isometry group.
By Proposition 1.2.6 we have that complex space forms are homogeneous spaces. It will
be interesting to represent them as a quotient of Lie groups.
Proposition 1.2.11 ([War71] Theorem 3.62 page 123). Let η : G × M → M be a transitive
action of the Lie group G over the manifold M. Let m 0 ∈ M and H the isotropy group of m 0 .
Then, the map
β : G/H −→ M
gH ↦→ η(g, m 0 )
is a diffeomorphism.

14 Spaces of constant holomorphic curvature
The Lie group P U ɛ (n) acts transitively over CK n (ɛ) and the isotropy group of a point in
CK n (ɛ), for each ɛ, is isomorphic to P (U(1) × U(n)), a closed Lie subgroup of P U ɛ (n). Thus,
we can represent CK n (ɛ) as a quotient of Lie groups
CK n (ɛ) ∼ = P U ɛ (n)/P (U(1) × U(n)) ∼ = U ɛ (n)/(U(1) × U(n)),
where the first diffeomorphism is given by x ↦→ g · P (U(1) × U(n)) with g ∈ P U ɛ (n) such that,
if x 0 ∈ CK n (ɛ) is fixed then g(x 0 ) = x.
Definition 1.2.12. A Riemannian manifold is a 2-point homogeneous space if the isometry
group of the manifold acts transitively in the unit tangent bundle.
Thus, complex space forms are 2-point homogeneous spaces. Real space forms are also 2-
point homogeneous spaces but, they are also 3-point homogeneous spaces, that is, the isometry
group acts transitively for triplets of a point and two orthonormal vectors in the tangent space
of the point. Complex space forms (ɛ ≠ 0) cannot be 3-point homogeneous spaces since the
sectional curvature is preserved by isometries and the sectional curvature is not constant.
1.3 Moving frames
Definition 1.3.1. Let U ⊂ M be an open set of a differentiable manifold. An orthonormal
moving frame of CK n (ɛ) defined at U is a map g 0 : U → CK n (ɛ) together with a collection of
g i : U → T CK n (ɛ) (i ∈ {1, . . . , 2n}) such that 〈g i , g j 〉 ɛ = δ ij where 〈 , 〉 ɛ denotes the Hermitian
product of CK n (ɛ) (defined at (1.4)) and π : T CK n (ɛ) → CK n (ɛ) is the canonical projection.
Definition 1.3.2. Let V be a 2n-dimensional real vector space endowed with a complex
structure J. An orthonormal basis {v 1 , v 2 , . . . , v 2n } is said to be a J-basis if v 2i = Jv 2i−1 for
every i ∈ {1, . . . , n}.
We denote by {e 1 , e 1 = Je 1 , . . . , e n , e n = Je n } the J-bases of V , and by {ω 1 , ω 1 , . . . , ω n , ω n }
the dual basis of a J-basis.
Remark 1.3.3. A J-basis is a special type of an orthonormal basis in a real vector space with
an almost complex structure J.
Definition 1.3.4. An orthonormal moving frame of CK n (ɛ) such that vectors {g 1 (p), . . . , g 2n (p)}
constitute a J-basis for all x ∈ U, is called a J-moving frame.
J-moving frames in CK n (ɛ) play an important role since they are in correspondence with
the elements of the isometry group of CK n (ɛ).
Consider F(CK n (ɛ)) the bundle of J-moving frames of CK n (ɛ), constituted by J-moving
frames (g 0 ; g 1 , Jg 1 , . . . , g n , Jg n ) with g 0 ∈ CK n (ɛ) and {g 1 , Jg 1 , . . . , g n , Jg n } a J-basis of
T g0 CK n (ɛ).
Proposition 1.3.5. The bundle of J-moving frames F(CK n (ɛ)) is identified with the isometry
group of CK n (ɛ).
Proof. We study the case ɛ ≠ 0. Let A ∈ U ɛ (n). By definition (1.5) of U ɛ (n) we have that A is
an (n + 1) × (n + 1) matrix with complex entries and such that its columns {a 0 , . . . , a n } satisfy
⎧
⎨
⎩
(a 0 , a 0 ) = sign(ɛ)1,
(a 0 , a i ) = 0, i ∈ {1, . . . , n},
(a i , a j ) = δ ij i, j ∈ {1, . . . , n},
where ( , ) denotes the Hermitian product in C n+1 defined at (1.2).
(1.9)

1.3 Moving frames 15
From the first property in the former list, we can take a 0 as a representative of g 0 = π(a 0 ) ∈
CK n (ɛ).
From the second property of (1.9) we have that a i satisfies the condition of being a vector
in T a0 H (cf. Section 1.2.2). Moreover, Re(Ja 0 , a i ) = Im(a 0 , a i ) = 0 and Re(Ja 0 , Ja i ) =
Re(a 0 , a i ) = 0. Let us consider g i := dπ(a i ) and Jg i := dπ(Ja i ) where π denotes the projection
defined at (1.3).
From the third condition in (1.9), vectors {g 1 , Jg 1 , . . . , g n , Jg n } constitute a J-basis of the
tangent space at g 0 .
Reciprocally, given a J-moving frame {g 0 ; g 1 , Jg 1 , . . . , g n , Jg n } defined on an open set, we
can define a matrix of U ɛ (n) (with the entries depending continuously on a parameter) just
taking as the first column the representative a 0 of g 0 with norm sign(ɛ)1. For the other columns
we consider the horizontal lift of g j at a 0 . As {g 1 , Jg 1 , . . . , g n , Jg n } is, in each point g, a J-basis
and we choose the horizontal lift, the columns of the constructed matrix verify the conditions
in (1.9) and are in U ɛ (n).
Definition 1.3.6. The unit tangent bundle of CK n (ɛ), denoted by S(CK n (ɛ)), is defined as
S(CK n (ɛ)) =
⋃
p∈CK n (ɛ)
T ′ pCK n (ɛ)
where T ′ pCK n (ɛ) denotes the sphere of unit vectors in the tangent vector space of CK n (ɛ) at
p.
In Lemma 1.2.7 we defined the invariant forms {ϕ ij } of U ɛ (n) as
ϕ ij (A; ·) = (da i (·), a j )
where A = (a 0 , . . . , a n ) ∈ U ɛ (n). As forms {ϕ ij } takes complex values we consider
ϕ jk = α jk + iβ jk (1.10)
Using the identification between U ɛ (n) and F(CK n (ɛ)), we can consider forms {ϕ ij } as
forms of F(CK n (ɛ)).
On the other hand, consider the canonical projections
and local sections
π 1
F(CK n (ɛ)) −→ S(CK n (ɛ)) −→ CK n (ɛ)
(g; g 1 , . . . , Jg n ) ↦→ (g, g 1 ) ↦→ g
CK n (ɛ) ⊃ U s 2
−→ S(CK n (ɛ)) ⊃ V s 1
−→ F(CK n (ɛ)).
Using the forms ϕ ij defined in F(CK n (ɛ)) and the previous local sections, we define the
following local invariant forms in S(CK n (ɛ))
and the local invariant forms in CK n (ɛ)
s ∗ 1(ϕ ij ), s ∗ 1(α ij ) and s ∗ 1(β ij )
s ∗ 2s ∗ 1(ϕ ij ), s ∗ 2s ∗ 1(α ij ) and s ∗ 2s ∗ 1(β ij ).
Lemma 1.3.7. Forms s ∗ 1 α 01, s ∗ 1 β 01 and s ∗ 1 β 11 are global forms in S(CK n (ɛ)).
π 2

16 Spaces of constant holomorphic curvature
Proof. If V ∈ T (p,v) (S(CK n (ɛ))) then
s ∗ 1α 01 (V ) (p,v) = Re((dπ 2 (V ), v)) = 〈dπ 2 (V ), v〉 ɛ
s ∗ 1β 01 (V ) (p,v) = Im((dπ 2 (V ), v)),
s ∗ 1β 11 (V ) (p,v) = Im((∇V, v)),
where ∇ denotes the Levi-Civita connection defined by ∇ : T (SCK n (ɛ)) → T CK n (ɛ), from
the Levi-Civita connection of CK n (ɛ). Inded, a vector V ∈ T (SCK n (ɛ)) is a tangent vector
to a curve of unit vectors, and in each of them we can apply the Levi-Civita connection of
CK n (ɛ).
We denote by α, β, γ the forms s ∗ 1 α 01, s ∗ 1 β 01, s ∗ 1 β 11, respectively.
Remark 1.3.8. The 1-form α coincides with the standard contact form of the unit tangent
bundle S(CK n (ɛ)).
On the other hand, forms s ∗ 2 s∗ 1 (ϕ ij) coincide with forms φ ij of CK n (ɛ) we define in the
following.
Let {g; g 1 , Jg 1 , . . . , g n , Jg n } be a J-moving frame on CK n (ɛ). As in the tangent space of
each point g, {g 1 , Jg 1 , . . . , g n , Jg n } defines a J-basis, we can consider the vectors {g 1 , . . . , g n }
as complex vectors. Then, the following differential forms are well-defined
φ j (·) = (dg(·), g j ) ɛ and φ jk (·) = (∇g j (·), g k ) ɛ (1.11)
where j, k ∈ {1, ..., n}, and ∇ denotes the Levi-Civita of CK n (ɛ) (i.e. we consider g j as a real
vector, we apply the Levi-Civita connection and we consider the result again as a complex
vector). Note that the differential forms φ j and φ jk are complex valued. We denote
φ j = α j + iβ j (1.12)
φ jk = α jk + iβ jk .
At Chapter 3, we work with orthonormal moving frames not necessarily J-moving frames.
Analogously, if {g; g 1 , g 2 , . . . , g 2n−1 , g 2n } is a moving frame on CK n (ɛ), we define the dual and
connection forms for this moving frame. We denote
ω j (·) = 〈dg(·), g j 〉 ɛ and ω jk (·) = 〈∇g j (·), g k 〉 ɛ (1.13)
with j, k ∈ {1, ..., 2n}, 〈 , 〉 ɛ the Hermitian product defined on CK n (ɛ) (see (1.4)), and ∇ the
Levi-Civita connection of CK n (ɛ).
Note that the differential forms {α j , β j } are a particular case of forms {ω j }: they are
obtained if we consider a J-moving frame.
Notation 1.3.9. Along this work we use invariant forms defined at CK n (ɛ), S(CK n (ɛ)) or
F(CK n (ɛ)), but we denote all of them by ϕ ij , α ij , β ij , without the pull-back of the sections,
if it is clear by the context.
Definition 1.3.10. Given a domain Ω ⊂ CK n (ɛ) we define the unit normal bundle of ∂Ω by
N(Ω)={(p, v) : p ∈ ∂Ω, v such that 〈v, w〉 ɛ ≥ 0 ∀ w tangent to a curve at Ω by p and ||v|| ɛ =1}.
Remark 1.3.11. The main results of this work, given at Chapters 3 and 4, have as a hypothesis
that the domain Ω ⊂ CK n (ɛ) which we take is compact with C 2 boundary. We denote by
regular domain a domain satisfying these hypothesis. We suppose that domains are regular in
order to simplify the arguments and to use techniques of differential geometry (for instance,
to have a well-defined second fundamental form in the whole boundary of the domain). These
hypothesis can be relaxed since most of the used results, mainly in valuations (see Chapter 2),
are known for a more general class of domains (cf. [Ale07a]).

1.4 Submanifolds 17
Lemma 1.3.12. Let Ω ⊂ CK n (ɛ) be a regular domain. Forms α and dα vanishes at N(Ω) ⊂
S(CK n (ɛ)).
Proof. Let V ∈ T (p,v) Ω. Then, α(V ) (p,N) = 〈dπ 2 (V ), N〉 ɛ = 0 since dπ 2 (V ) is a tangent vector
tangent at ∂Ω at point p.
In order to prove that the 2-form dα vanishes at the unit normal bundle, we consider the
inclusion of the unit normal bundle to the unit tangent bundle i : N(Ω) → S(Ω). Using that
the differential map commutes with the inclusion we have the result
dα| N(Ω) = (i ∗ ◦ d)(α) = (d ◦ i ∗ )(α) = d(i ∗ α) = d(0) = 0.
1.4 Submanifolds
1.4.1 Totally geodesic submanifolds
Definition 1.4.1. Let M be a Riemannian manifold. A submanifold N ⊂ M is totally geodesic
if every geodesic in the submanifold N is also a geodesic in M.
As C n is metrically equivalent to R 2n , totally geodesic submanifolds in C n coincide with
the ones in R 2n . For the other complex space forms, the totally geodesic submanifolds are
classified.
Definition 1.4.2. Let V be a real vector space of dimension 2n endowed with an almost complex
structure J compatible with a scalar product 〈 , 〉. It is said that vectors {e 1 , . . . , e m } expand
a complex subspace if the space generated by these vectors is J-invariant, i.e. J(span{e 1 , . . . , e m }) =
span{e 1 , . . . , e m }.
It is said that a submanifold of a complex manifold is complex if at each point, the tangent
space of the submanifold is complex subspace of the tangent space of the manifold.
Definition 1.4.3. Let V be a real vector space of dimension 2n endowed with an almost
complex structure J compatible with a scalar product 〈 , 〉. It is said that vectors {e 1 , . . . , e m }
expand a totally real subspace if
〈e i , Je j 〉 = 0, ∀i, j ∈ {1, . . . , m}.
It is said that a submanifold of a complex manifold is totally real if at each point, the tangent
space of the submanifold is a totally real subspace of the tangent space of the manifold.
Theorem 1.4.4 ([Gol99] pages 75 and 80). Let z ∈ CK n (ɛ).
1. If L ⊂ T z CK n (ɛ) is a complex vector subspace with complex dimension r, then there
exists a unique complete complex totally geodesic submanifold through z and tangent to
L at z.
2. If L ⊂ T z CK n (ɛ) is a totally real vector space of real dimension k, then there exists a
unique complete totally geodesic totally real submanifold through z and tangent to L at
z.
Definition 1.4.5. The complex submanifold defined at 1. in the previous theorem is called
complex r-plane, and denoted by L r .
The totally real submanifold defined at 2. in the previous theorem is called totally real
k-plane, and denoted by L R k .

18 Spaces of constant holomorphic curvature
In the projective model, complex r-planes are obtained from the projection of a subspace
F ⊂ C n+1 intersection CK n (ɛ). The subspace F is the (r + 1)-dimensional complex vector
subspace spanned by a representative z ′ of z = π(z ′ ) and by the horizontal lift of vectors in
L ⊂ T z CK n (ɛ) (cf. [Gol99, Section 3.1.4]).
Analogously, totally real k-planes are obtained from the projection of F R ⊂ C n+1 intersection
CK n (ɛ), where F R is the (k + 1)-dimensional real vector subspace spanned by a
representative z ′ of z = π(z ′ ) and by the horizontal lift of vectors in L ⊂ T z CK n (ɛ).
Theorem 1.4.6 ([Gol99] p. 82). The unique complete totally geodesic submanifolds in CK n (ɛ)
are the complex r-planes, r ∈ {1, . . . , n − 1} and the totally real k-planes, k ∈ {1, . . . , n}.
Corollary 1.4.7. In CK n (ɛ), ɛ ≠ 0, there are not totally geodesic (real) hypersurfaces.
That is, it does not exist the equivalent hypersurface to a hyperplane in a real space form.
The more reasonable substitutes of hyperplanes are the so-called bisectors, which we study on
page 82.
Theorem 1.4.6 will be important along this work since it will be interesting to know which
totally geodesic submanifolds can be taken in a complex space form as a substitutes of (totally
geodesic) planes in real space forms.
1.4.2 Geodesic balls
A geodesic ball in a Riemannian manifold is the set of points equidistant from a fixed point
called center.
In real space forms, geodesic balls are totally umbilical real hypersurfaces, i.e. the second
fundamental form is, at every point, a multiple of the identity and the same multiple for every
point.
This fact does not hold in complex projective and hyperbolic spaces. Moreover, in these
spaces, there are no totally umbilical real hypersurface.
Proposition 1.4.8 ([Mon85]). The principal curvatures of a sphere of radius r in CK n (ɛ),
ɛ ≠ 0 are
i) 2 cot ɛ (2r) with multiplicity 1 and principal direction −JN (where N denotes the inward
normal vector to the sphere),
ii) cot ɛ (r) with multiplicity 2n − 2.
Recall that cos ɛ , sin ɛ denote the generalized trigonometric functions defined at Notation
1.2.2.
Along this work, we use the value of the mean curvature integrals for a geodesic ball of
radius R in CK n (ɛ) (cf. Definition 2.1.4).
From the previous proposition we have that the symmetric elementary functions are
⎧
⎪⎨
⎪⎩
σ 0 = 1
) −1
( (2n−2
σ i = ( 2n−1
i
σ 2n−1 = 2 cot 2n−2
ɛ
i
)
cot
i
ɛ (R) + ( 2n−2
i−1
(R) cot ɛ (2R).
) )
2 cot
i−1(R) cot ɛ (2R)
By a straightforward computation, we obtain that the expression of the mean curvature integrals
is
and
⎧
⎪⎨
⎪⎩
M 0 = vol(∂B R ) =
2π
M i =
n
(2n−1)(n−1)!
M 2n−1 =
2πn
(n−1)! (cos2n ɛ
2πn
(n−1)! sin2n−1 ɛ
((2n + i − 1) cosi+1 ɛ
(R) + cos 2n−2
ɛ
(R) cos ɛ (R)
(R) sin 2n−i−1
ɛ
(R) sin 2 ɛ(R))
vol(B R ) = πn
n! (sin ɛ(R)) 2n .
ɛ
(R) − i cos i−1 (R) sin 2n−i−1 (R))
ɛ
ɛ

1.5 Space of complex r-planes 19
1.5 Space of complex r-planes
We denote the space of all complex r-planes in CK n (ɛ) by L C r and the space of all totally real
r-planes in CK n (ɛ) by L R r (cf. Definition 1.4.5).
We shall use that L C r is a homogeneous space with respect to the isometry group of CK n (ɛ).
In order to prove this fact, we need the following result.
Lemma 1.5.1. The group of isometries of CK n (ɛ) acts transitively on the J-bases.
Proof. We study the case ɛ ≠ 0. Fix the canonical J-basis {e 1 , Je 1 , . . . , e n , Je n } at e 0 ∈
CK n (ɛ). It is enough to prove that given another J-basis {g 1 , Jg 1 , . . . , g n , Jg n } of T g0 CK n (ɛ),
there exists an isometry ρ which takes this J-basis to the fixed one.
Take as isometry ρ ∈ U ɛ (n) the matrix with columns (˜g 0 , ˜g 1 , . . . , ˜g n ) where ˜g 0 is a representative
of g 0 with norm ɛ and ˜g i is the horizontal lift of g i at ˜g 0 . In the same way as in the
proof of Proposition 1.3.5 we have that ρ is a matrix in U ɛ (n). Moreover, it carries the fixed
J-basis to the given one.
From the previous lemma we get
Lemma 1.5.2. The space of complex r-planes is a homogeneous space with respect to the
isometry group of CK n (ɛ).
Proof. We define a J-basis {e 1 , Je 1 , . . . , e r , Je r } of the tangent space in any point of a complex
r-plane. Completing this J-basis to a J-basis of the whole space CK n (ɛ), and applying the
previous lemma we get the result.
In order to study integral geometry in CK n (ɛ), it is necessary that the space L C r admits an
invariant density under the isometries of CK n (ɛ). In general, the absolute value of a form of
maximum degree is called a density. By the following lemma, it is enough to prove that L C r is
the quotient of unimodular groups.
Lemma 1.5.3 ([San04]). If G, H are unimodular groups, then G/H admits an invariant
density.
The isotropy group of a complex r-plane is isomorphic to
U ɛ (r) × U(n − r) =
{
( A 0
X ∈ M (n+1)×(n+1) (C) : X =
0 B
)
}
, A ∈ U ɛ (r), B ∈ U(n − r)
(1.14)
since these matrices (and only these) leave invariant a complex r-plane and its orthogonal.
Then,
L C r ∼ = U ɛ (n)/(U ɛ (r) × U(n − r)).
The group U ɛ (n), by Lemma 1.2.9, is unimodular, and U ɛ (r)×U(n−r) is also a unimodular
group. Thus, there exists an invariant density in the quotient space, that is, in the space of
complex r-planes.
The following result give a method to obtain explicitly the density in the quotient space.
Theorem 1.5.4 ([San04] page 147). Let G be a Lie group with dimension n and H a closed
subgroup of G with dimension n − m. Then, G/H is a homogeneous space with dimension m.
Let ˜ω be the m-form obtained from the product of all invariant 1-forms of G such that they
vanish on H. Then, there exists an invariant density ω in G/H if and only if d˜ω = 0. In
this case, ˜ω is the pull-back of ω for the canonical projection of G at G/H (up to constants
factors).

20 Spaces of constant holomorphic curvature
Proposition 1.5.5. Let π : U ɛ (n) −→ U ɛ (n)/(U ɛ (r) × U(n − r)) ∼ = L C r . If dL r is an invariant
density of L C r then
∧
π ∗ dL r = ϕ ij ∧ ϕ ij
i=0,...,r
j=r+1,...,n
or, equivalently,
π ∗ dL r =
∧
i=0,...,r
j=r+1,...,n
α ij ∧ β ij ,
where {ϕ ij , α ij , β ij } are the forms defined at Lemma 1.2.7 and at (1.11).
Proof. The result is known for C n , see [San04].
For the other complex space forms, we note that the (real) dimension of the space of
complex r-plans coincides with the degree of π ∗ dL r (and it is equal to 2(r + 1)(n − r)). Thus,
it suffices to prove that each form ϕ ij with i ∈ {0, . . . , r}, j ∈ {r + 1, . . . , n} vanishes over
U ɛ (n) × U(n − r). But, from the form of matrices in U ɛ (n) × U(n − r) given at (1.14), the
result follows immediately.
Let us give an example of moving frames in the space of complex r-planes using Definition
1.3.1, which will be used in the next section.
Let us take as the open set U ⊂ M (see Definition 1.3.1) an open set in L C r . An orthonormal
frame is given by
g : U ⊂ L C r −→ CK n (ɛ)
L r ↦→ p ∈ L r
and
g i : U ⊂ L C r −→ T CK n (ɛ)
L r ↦→ v i ∈ T g(Lr)L r
, (1.15)
i ∈ {1, . . . , 2r}, such that 〈v i , v j 〉 ɛ = δ ij . It will be interesting to consider that Jg 2k−1 (L r ) =
g 2k (L r ), k ∈ {1, . . . , r}, that is, vectors {g 1 (L r ), . . . , g 2r (L r )} constitute a J-basis at g(L r ). By
abuse of notation, we denote g(L r ) by g and g i (L r ) by g i .
Remark 1.5.6. From the correspondence between U ɛ (n) and the bundle of J-moving frames
F(CK n (ɛ)), and between U ɛ (n)/(U ɛ (r)×U(n−r)) and L C r we have that {p; g 1 (L r ), . . . , g 2r (L r )}
are sections of U ɛ (n) → U ɛ (n)/(U ɛ (r) × U(n − r)).
1.5.1 Expression for the invariant density in terms of a parametrization
Sometimes it is interesting to have a more geometric expression for the invariant density of
complex r-planes. For example, Santaló proved
Proposition 1.5.7 ([San04]). The invariant density of the space of totally geodesic planes in
a space form of constant sectional curvature k is
dL r = cos r k (ρ)dx n−r ∧ dL (n−r)[O] (1.16)
where dx n−r is the volume element of the (n − r)-plane orthogonal to L r containing the origin
O and dL (n−r)[O] is the volume element of the Grassmannian of (n − r)-planes containing the
origin.
Now, we give a similar expression in complex space forms, for the density of complex
r-planes in CK n (ɛ).

1.5 Space of complex r-planes 21
Proposition 1.5.8. The invariant density of the space of complex r-planes in a space form
of constant holomorphic curvature 4ɛ is
dL r = cos 2r
ɛ (ρ)dx n−r ∧ dL (n−r)[O]
where dx n−r is the volume element of the complex (n−r)-plane orthogonal to L r containing the
origin O and dL (n−r)[O] is the volume element of the Grassmannian of complex (n − r)-planes
containing the origin.
Proof. In order to obtain the expression of the invariant density of L C r in terms of dL (n−r)[O]
and dx n−r , we follow the same idea as in [San04] (where it is proved for the Euclidean and the
real hyperbolic space). That is, we fix an adapted J-moving frame to the complex r-plane,
defined in a neighborhood of the point at minimum distance from the origin O, then, by
parallel translation, we translate this moving frame to the origin O. Finally, we relate both
moving frames from the pull-back of a section.
Denote by G = U ɛ (n) and by H = U ɛ (r)×U(n−r) the isotropy group of a complex r-plane.
The projection π : G −→ G/H gives, with respect to a J-moving frame adapted to the
complex r-plane and the forms defined at Lemma 1.2.7,
π ∗ dL r =
n∧
j=r+1
α j0 ∧ β j0
∧
i=1,...,r
j=r+1,...,n
α ji ∧ β ji .
Let O ∈ CK n (ɛ) be the fixed origin and let L r ∈ L C r . We denote by p(L r ) the point in L r
at a minimum distance from O.
Let i be a local section of π. Then, π ◦ i = id and i ∗ π ∗ dL r = dL r , so that, we can obtain
the expression of dL r . From the identifications explained in Remark 1.5.6, we take as a section
i the defined by {p(L r ); g 1 , Jg 1 , . . . , g n , Jg n }, a J-moving frame defined in a neighborhood V
of L r , adapted to L r and such that g r+1 is the tangent vector to the geodesic joining p(L r )
and O. Denote by {g 1 , Jg 1 , . . . , g n , Jg n } the dual basis of {g 1 , Jg 1 , . . . , g n , Jg n } at g 0 . From
Proposition 1.3.5, we consider the matrix in G corresponding to this J-moving frame. Denote
the columns of G also by (g 0 , g 1 , . . . , g n ), so that (g i ◦ i) denotes the i-th column of the matrix.
Then, using the same notation as in (1.12), we have
n∧
i ∗ ( α j0 ∧ β j0 ) =
j=r+1
=
n∏
n∏
α j0 (di) ∧ β j0 (di) = 〈dg 0 (di), g j 〉〈dg 0 (di), Jg j 〉
j=r+1
n∏
j=r+1
〈d(g 0 ◦ i), g j 〉〈d(g 0 ◦ i), Jg j 〉 = (g 0 ◦ i) ∗ (
j=r+1
j=r+1
n∧
g j ∧ Jg j )
but (g 0 ◦ i) = p(L r ) and the previous form coincides with the volume element of p in the
subspace generated by {g r+1 , Jg r+1 , . . . , g n , Jg n }, which is a complex (n − r)-plane. Thus,
i ∗ (
n∧
j=r+1
α j0 ∧ β j0 ) = dx n−r .
Let G ′ ⊂ G be the subgroup of all J-moving frames of G such that g r+1 is the tangent
vector to the geodesic containing O and g 0 , and let G 0 ⊂ G be the subgroup of all J-moving
frames with base point O. Let ρ be the distance from L r to O. Denote by s ρ the parallel
translation from O to p(L r ) along the geodesic. Consider the following maps
π 1 : G ′ −→ G 0
(g 0 ; g 1 , . . . , g n ) ↦→ (O; s −1
ρ (g 1 ), . . . , s −1
ρ (g n )) ,

22 Spaces of constant holomorphic curvature
π 2 : G 0 −→ L C r[O]
G 0 ↦→ complex r-plane containing O generated by g 1 , . . . , g r
,
π 3 : L C r −→ L C r[0]
L r ↦→ ((L r ) ⊥ O )⊥ O
where (L r ) ⊥ O denotes the orthogonal space to L r by O. Then, the following diagram is commutative
G ′ π −−−−→
1
G0
⏐ ⏐
π↓
↓ π 2
(1.17)
L C r
We define curves x ij in G 0 ⊂ G as follows
π 3
−−−−→ L
C
r[0]
.
x ij (t) = (O; g 1 , . . . , g i (t), Jg i (t), . . . , g j (t), Jg j (t) . . . , g n , Jg n )
where g i (t) = cos(t)g i + sin(t)g j and g j (t) = − sin(t)g i + cos(t)g j and curves
x j i (t) = (O; g 1, . . . , g i (t), Jg i (t), . . . , g j (t), Jg j (t), . . . , g n , Jg n )
where g i (t) = cos(t)g i + sin(t)Jg j and g j (t) = − sin(t)Jg i − cos(t)g j .
From the local section i, we have
i ∗ α ji = i ∗ ((π ∗ 1 ◦ s ∗ )(α ji )) = (π 1 ◦ i) ∗ (s ∗ ρα ji ),
i ∗ β ji = i ∗ ((π ∗ 1 ◦ s ∗ )(β ji )) = (π 1 ◦ i) ∗ (s ∗ ρβ ji ).
Thus, we have to study (s ∗ ρα ij )(ẋ kl ) and (s ∗ ρα ij )(ẋ l k ) (and the same for β ij). Then, we need
(g l ◦ s ρ )(x ij ) since
Note that
(s ∗ ρα ij )(ẋ kl ) = α ij (ds ρ (ẋ kl )) = 〈g j
∣
∣sρ(x
kl (t)) , d(g i ◦ s ρ ) ∣ ∣
sρ(x kl (t)) (ẋ kl)〉. (1.18)
(g i ◦ s ρ )(x kl ) = i-th column of the matrix s ρ (x kl )
and that s ρ (x kl ) ∈ G ′ is obtained from the parallel translation along the geodesic for O and
with tangent vector g r+1 (t) in O. Hence, for curves x ij , x j i
with i, j ≠ r + 1 we always take
parallel translation along the same geodesic, when we apply s ρ .
When we move g r+1 (t) we consider the parallel translation along different geodesics for each
t. But, as curves x r+1,j , x j r+1 , x 1,r+1 just move the vector g r+1 (t) in a real plane generated by
{g r+1 (0), g j (0)} or {g r+1 (0), Jg j (0)}, we have that g 0 (s ρ (x r+1,j )(t)) or g 0 (s ρ (x j r+1 (t))) describes
a circle in CK n (ɛ) contained in the plane generated by {g r+1 (0), g j (0)} (or {g r+1 (0), Jg j (0)}).
From these remarks we have
• (g 0 ◦ s ρ )(x kl ): point at a distance ρ from O in which we arrive along the geodesic with
tangent vector g r+1 at O.
⋄ x r+1,l , l > r + 1.
⋄ x l r+1 , l ≥ r + 1.
(g 0 ◦ s ρ )(x r+1,l ) = cos ɛ (ρ)O + sin ɛ (ρ)(cos(t)g r+1 + sin(t)g l ).
(g 0 ◦ s ρ )(x l r+1) = cos ɛ (ρ)O + sin ɛ (ρ)(cos(t)g r+1 + sin(t)(Jg l )).
,

1.5 Space of complex r-planes 23
⋄ x 1,r+1 .
(g 0 ◦ s ρ )(x 1,r+1 ) = cos ɛ (ρ)O + sin ɛ (ρ)(− sin(t)g 1 + cos(t)g r+1 )
since g 1 (t) = g 1 cos(t) + g r+1 sin(t) and g r+1 (t) is perpendicular to g 1 (t) for all t.
Then, g r+1 (t) = −g 1 sin(t) + g r+1 cos(t).
⋄ x r+1
1 .
⋄ x kl , k < l, k, l ≠ r + 1.
(g 0 ◦ s ρ )(x r+1
1 ) = cos ɛ (ρ)O + sin ɛ (ρ)(sin(t)(Jg 1 ) + cos(t)g r+1 ).
(g 0 ◦ s ρ )(x kl ) = cos ɛ (ρ)O + sin ɛ (ρ)g r+1 .
⋄ x l k , k ≤ l, k, l ≠ r + 1. (g 0 ◦ s ρ )(x l k ) = cos ɛ(ρ)O + sin ɛ (ρ)g r+1 .
• (g r+1 ◦ s ρ )(x kl ):
⋄ x r+1,l , l > r + 1.
(g r+1 ◦ s ρ )(x r+1,l ) = s −1
ρ (cos(t)g r+1 + sin(t)g l )
but this parallel translation coincides with the tangent vector to the geodesic at
time ρ, that is,
s −1
ρ (cos(t)g r+1 +sin(t)g l ) = tangent vector to (cos ɛ (ρ)O+sin ɛ (ρ)(cos(t)g r+1 +sin(t)g l ))
= sin ɛ (ρ)O + cos ɛ (ρ)(cos(t)g r+1 + sin(t)g l ).
⋄ x l r+1 , l ≥ r + 1.
⋄ x 1,r+1 .
⋄ x r+1
1 .
⋄ x kl , k < l, k, l ≠ j.
(g r+1 ◦ s ρ )(x l r+1) = sin ɛ (ρ)O + cos ɛ (ρ)(cos(t)g r+1 + sin(t)(Jg l )).
(g r+1 ◦ s ρ )(x 1,r+1 ) = sin ɛ (ρ)O + cos ɛ (ρ)(− sin(t)g 1 + cos(t)g r+1 ).
(g r+1 ◦ s ρ )(x r+1
1 ) = sin ɛ (ρ)O + cos ɛ (ρ)(sin(t)(Jg 1 ) + cos(t)g r+1 ).
(g r+1 ◦ s ρ )(x kl ) = sin ɛ (ρ)O + cos ɛ (ρ)g r+1 .
⋄ x l k , k ≤ l, k, l ≠ j. (g r+1 ◦ s ρ )(x l k ) = sin ɛ(ρ)O + cos ɛ (ρ)g r+1 .
• (g j ◦ s ρ )(x kl ), j > r + 1:
⋄ x jl , l > j.
(g j ◦ s ρ )(x jl ) = s −1
ρ (cos(t)g j + sin(t)g l ).

24 Spaces of constant holomorphic curvature
⋄ x l j , l ≥ j.
⋄ x lj , l < j.
⋄ x j l , l ≤ j.
⋄ x kl , k < l, k, l ≠ j.
⋄ x l k
, k ≤ l, k, l ≠ j.
(g
(g j ◦ s ρ )(x l j) = s −1
ρ (cos(t)g j + sin(t)(Jg l )).
(g j ◦ s ρ )(x lj ) = s −1
ρ (− sin(t)g l + cos(t)g j ).
(g j ◦ s ρ )(x j l ) = s−1 ρ (sin(t)(Jg l ) + cos(t)g j ).
(g j ◦ s ρ )(x kl ) = s −1
ρ (g j ).
j ◦ s ρ )(x l k ) = s−1 ρ (g j ).
Now, we compute s ∗ ρα ij (and s ∗ ρβ ij ) in terms of α ij , β ij using (1.18) and evaluating s ∗ ρα ij
(and s ∗ ρβ ij ) to each curve x kl . Doing so, we obtain
(s ∗ ρα j1 ) g = −(α j1 ) g ′, j > r + 1.
(s ∗ ρα r+1,1 ) g = − cos ɛ (ρ)(α r+1,1 ) g ′.
(s ∗ ρβ j1 ) g = (β j1 ) g ′, j > r + 1.
(s ∗ ρα r+1,1 ) g = − cos ɛ (ρ)(β r+1,1 ) g ′.
Finally, we have
s ∗ ρ(
∧
j=r+1,...,n
i=1,...,r
α ji ∧ β j1 ) g = cos 2r
ɛ (ρ)(
∧
j=r+1,...,n
i=1,...,r
α j1 ∧ β j1 ) g ′. (1.19)
To get an expression for
i ∗ (
∧
j=r+1,...,n
i=1,...,r
α ji ∧ β j1 )
we use the diagram (1.17) and the computation in (1.19), so that
i ∗ (
∧
j=r+1,...,n
i=1,...,r
α ji ∧ β ji ) g = i ∗ ◦ π ∗ 1 ◦ s ∗ ρ(
∧
j=r+1,...,n
i=1,...,r
= i ∗ ◦ π ∗ 1(cos 2r
ɛ (ρ)(
∧
j=r+1,...,n
i=1,...,r
α ji ∧ β ji ) g
α ji ∧ β ji ) ′ g)
= cos 2r
ɛ (ρ)π ∗ 3(dL r[0] ) = cos 2r
ɛ (ρ)dL (n−r)[0]
where we used the expression for the invariant density of complex r-planes through a point,
cf. (1.20), and the duality between complex r-planes through a point and the complex (n − r)-
planes through the same point.
Hence, we get
dL r = cos 2r
ɛ (ρ)dx n−r dL (n−r)[O] .

1.5 Space of complex r-planes 25
1.5.2 Density of complex r-planes containing a fixed complex q-plane
We denote by L C r[q]
the space of complex r-planes containing a fixed complex q-plane.
We denote by H [q] := U ɛ (q) the isotropy group of a complex q-plane in CK n (ɛ) and by H r[q]
the isotropy group of a complex r-plane containing the fixed complex q-plane. Note that H [q]
acts transitively on L C r[q] .
On the other hand, if we suppose that L 0 q is the fixed complex q-plane, then we can define
the projection
π : H [q] −→ H [q] /H r[q]
g ↦→ L r[q] = gL 0 q.
As the elements in H r[q] do not mix tangent vectors to the fixed complex q-plane with
orthogonal vectors to this complex q-plane, we have
⎧⎛
⎞
⎫
⎨ A 0 0
⎬
H r[q]
∼ = ⎝ 0 B 0 ⎠ , A ∈ U ɛ (q), B ∈ U(r − q), C ∈ U(n − r)
⎩
⎭ ,
0 0 C
π ∗ dL r[q] =
∧
q+1≤i≤r
r+1≤j≤n
α ij ∧ β ij . (1.20)
(The forms vanishing, with respect to the isometry group, in this case H [q] , are the ones
inside the big box.)
1.5.3 Density of complex q-planes contained in a fixed complex r-plane
Denote by L (r)
q the space of complex q-planes contained in a fixed complex r-plane. Let us
fix a complex r-plane L r and a complex q-plane L (r)
q contained in L r . Consider the projection
π : U ɛ (r) → U ɛ (r)/U ɛ (q) × U(r − q) where U ɛ (r) denotes the isometry group of the fixed
complex r-plane and U ɛ (q) × U(r − q) the isotropy group of a complex q-plane contained in
the complex r-plane. Then, as in the previous case we get
π ∗ dL (r)
q = ∧
1≤i≤q
q+1≤j≤r
α ij ∧ β ij
∧
q+1≤j≤r
α j0 ∧ β j0 =
∧
0≤i≤q
q+1≤j≤r
α ij ∧ β ij .
1.5.4 Measure of complex r-planes intersecting a geodesic ball
In order to obtain the value of this measure we use the expression for the invariant density
of complex r-planes in (1.16) and the expression of the Jacobian of the map of changing to
spherical coordinates. This is given by (cf. [Gra73])
cos ɛ (R) sinɛ
2n−1 (R)
|ɛ| (n−1)/2 .
Recall that cos ɛ and sin ɛ denote the generalized trigonometric functions defined at Notation
1.2.2.
We fix as a origin of the spherical coordinates the center of the geodesic ball. Then, using
the expression in spherical coordinates for the element volume of CK n−r (ɛ) (the orthogonal

26 Spaces of constant holomorphic curvature
space to a complex r-plane intersecting the sphere), we obtain
∫
∫ ∫
m(L r ∈ L C r : B R ∩ L r ≠ ∅) = dL r =
cos 2r
ɛ (ρ)dx n−r ∧ dG C
B R ∩L r≠∅
G C n,n−r[O]
n,n−r B R ∩L r
∫
= vol(GC n,n−r)
|ɛ| (n−1)/2
cos 2r+1
ɛ (ρ) sin 2(n−r)−1
ɛ (ρ)dρdS 2(n−r)−1
∫ R
S 2(n−r)−1 0
= vol(GC n,n−r)
|ɛ| (n−1)/2 vol(S2(n−r)−1 )
= vol(GC n,n−r)
|ɛ| (n−1)/2 vol(S2(n−r)−1 )
= vol(GC n,n−r)
|ɛ| (n−1)/2 vol(S2(n−r)−1 )
= vol(GC n,n−r)
|ɛ| (n−1)/2 vol(S2(n−r)−1 )
∫ R
0
∫ R
0
cos 2r+1
ɛ
(ρ) sinɛ
2(n−r)−1 (ρ)dρ
cos ɛ (ρ)(1 + sin 2 ɛ(ρ)) r sin 2(n−r)−1
ɛ (ρ)
r∑
( ∫ r R
i)
i=0
r∑
i=0
0
cos ɛ (ρ) sinɛ
2(n−r+i)−1 (ρ)
( 2(n−r+i)
r sin ɛ (R)
i)
2(n − r + i) .
At Chapter 4 we give a general expression for the measure of complex r-planes intersecting
a regular domain, in a way such that the previous expression can be interpreted in terms of
mean curvature integrals and other valuations defined at Chapter 2.
1.5.5 Reproductive property of Quermassintegrale
At Chapter 3 we prove that the mean curvature integral does not satisfy a reproductive
property (see Definition 3.4.1). In this section we prove that Quermassintegrale do satisfy a
reproductive property.
Definition 1.5.9. Let Ω be a domain in CK n (ɛ). For r ∈ {1, . . . , n − 1} we define
W r (Ω) = (n − r) · O ∫
r−1 · · · O 0
χ(Ω ∩ L r )dL r
n · O n−2 · · · O n−r−1
where O i denotes the area of the sphere of radius in the standard Euclidean space. Moreover,
we define
W 0 (Ω) = vol(Ω) and W n (Ω) = O n−1
n
χ(Ω).
Constants are chosen for analogy to the case of real space forms.
Proposition 1.5.10. Let Ω be a domain in CK n (ɛ). Then,
∫
W r (Ω) = c W r (Ω ∩ L q )dL q
for 1 ≤ r ≤ q ≤ n − 1 and c is a constant depending only on n, r and q.
L C q
Proof. This proof is analogous to the one given by Santaló (cf. [San04]) to obtain the result
in the Euclidean space.
By definition, it is satisfied
∫
L C q
W r (Ω ∩ L q )dL q = (q − r)O r−1 . . . O 0
qO q−1 . . . O q−r−1
L C r
∫
L C q
∫
Ω∩L r
(q) ≠∅
dL (q)
r ∧ dL q .
We express the densities dL (q)
r , dL q , dL q[r] , dL r in terms of the forms ϕ ij defined at Lemma
1.2.7 (we omit the absolute value for the densities),

1.5 Space of complex r-planes 27
and
Thus, the equality
dL (q)
r ∧ dL q = ∧
dL q[r] ∧ dL r =
holds.
Applying it we get the result
∫ ∫
L C q
Ω∩L (q)
r ≠∅
dL q =
∧
q

Chapter 2
Introduction to valuations
The notion of valuation in R n was introduced by Blaschke in 1955 at [Bla55]. Recently,
Alesker, among others, has extended this notion to differentiable manifolds. A survey about
the development of valuations is given at [MS83] where some references are given.
In this chapter we briefly introduce the theory of valuations, focusing on the concepts and
results we shall use in the following chapters.
In the last section, we define some valuations in the spaces of constant holomorphic curvature,
generalizing the definition of some valuations in C n . We also give relations among the
defined valuations.
2.1 Definition and basic properties
Let V be a vector space of real dimension n. We denote by K(V ) the set of non-empty compact
convex domains in V .
Definition 2.1.1. A functional φ : K(V ) → R is called a valuation if
whenever A, B, A ∪ B ∈ K(V ).
φ(A ∪ B) = φ(A) + φ(B) − φ(A ∩ B)
Remark 2.1.2. The extension theorem of Groemer states that every valuation extends uniquely
to the set of finite union of convex set.
First examples of valuations in R n are the volume of a convex domain, the area of its
boundary and its Euler characteristic. Intrinsic volumes, defined by the coefficients of the
polynomial obtained in the Steiner’s formula, are also valuations.
Consider in R n a convex domain Ω and denote by Ω r the parallel domain at a distance r
from Ω. Recall that the parallel domain is constituted by all points at a distance less or equal
than r.
The Steiner formula relates the volume of the parallel domain with the volume and some
other functionals of the initial domain.
Proposition 2.1.3 (Steiner’s formula). Let Ω ⊂ R n be a compact domain and Ω r the parallel
domain at distance r. Then, the volume of Ω r can be expressed as a polynomial in r and
its coefficients are multiples of the valuations V i : K(V ) → R, i ∈ {0, . . . , n}, called intrinsic
volumes. The explicit expression is
n∑
vol(Ω r ) = r n−i ω n−i V i (Ω) (2.1)
i=0
where ω n−i denotes the volume of the (n − i)-dimensional ball of radius 1 in the Euclidean
space.
29

30 Introduction to valuations
Proof. (of the fact that V i are valuations.)
Let A, B, A ∪ B ∈ K(V ). Then, it is satisfied (A ∩ B) r = A r ∩ B r and (A ∪ B) r = A r ∪ B r .
Thus,
vol((A ∪ B) r ) = vol(A r ) + vol(B r ) − vol(A r ∩ B r ) = vol(A r ) + vol(B r ) − vol((A ∩ B) r )
∀r.
Applying (2.1) we deduce that the intrinsic volumes are valuations.
Some particular cases of intrinsic volumes are
• V n (Ω) = vol(Ω),
• V n−1 (Ω) = vol(∂Ω)/2,
• V 0 (Ω) = χ(Ω).
Note that the Steiner formula has sense for any convex domain, without any assumption
on the regularity of the boundary. In some cases it is interesting to consider convex domains
such that its boundary is an oriented hypersurface of class C 2 . Applying the previous formula
in this case we obtain the so-called mean curvature integrals.
Definition 2.1.4. Let S be a hypersurface of class C 2 in a Riemannian manifold M of dimension
n. If x ∈ S, we denote the second fundamental form of S at x by II x . We define the i-th
mean curvature integral of S as
( ) n − 1 −1 ∫
M i (S) =
i
S
σ i (II x )dx
where σ i (II x ) denotes the r-th symmetric elementary function of the second fundamental form
II x .
Then, the Steiner formula in R n is
n−1
(
∑ n
)
vol(Ω r ) = vol(Ω) + r n−i i
n M n−i−1(∂Ω).
i=0
Sometimes, it is defined M −1 (∂Ω) := vol(Ω).
The relation among the intrinsic volumes and the mean curvature integrals, for convex
domains with boundary of class C 2 , is
( n
i)
V i (Ω) = M n−i−1 (∂Ω).
nω n−i
Definition 2.1.5. A valuation φ is continuous if it is continuous with respect to the Hausdorff
metric.
Recall that the Hausdorff distance between two compact sets A, B is given by
d Haus (A, B) = max{sup inf {d(a, b)}, sup inf {d(a, b)}}
a∈A b∈B
b∈B a∈A
where d(a, b) is the distance defined in the ambient space for A and B.
Example 2.1.6. The intrinsic volumes are continuous valuations. Anyway, there are some
interesting examples of non-continuous valuations in R n . For example, the affine surface area
is a valuation in the Euclidean space, but it is not continuous (cf. [KR97]). The affine surface
area of a convex domain Ω ⊂ R n is defined as the integral of the (n + 1)-th root of the

2.1 Definition and basic properties 31
generalized Gauss curvature (cf. [Sch93] notes of Sections 1.5 and 2.5) of the boundary of the
domain with respect to the (n − 1)-dimensional Hausdorff measure of the boundary
∫
n+1
AS(Ω) =
√ K x dx.
∂Ω
One of the most important property of this valuation is that it is invariant under translations
and linear transformations with determinant 1.
Definition 2.1.7. Given a (2n − 1)-form ω defined on S(V ), and a smooth measure, η we
consider, for each regular domain Ω,
∫ ∫
η + ω
N(Ω)
Ω
where N(Ω) denotes the normal bundle of the boundary of the domain. The obtained functional
is called smooth valuation.
Definition 2.1.8. Let Ω ∈ K(V ). A valuation φ : K(V ) → R is called
• translation invariant if
φ(ψΩ) = φ(Ω)
for every ψ translation of the vector space V ;
• invariant with respect to a group G acting on V if
for every g ∈ G;
• homogeneous of degree k if
φ(gΩ) = φ(Ω)
φ(λΩ) = λ k φ(Ω) for every λ > 0, k ∈ R;
• even (resp. odd) if
with ɛ even (resp. odd);
φ(−1 · Ω) = (−1) ɛ φ(Ω)
• monotone if
φ(Ω 1 ) ≥ φ(Ω 2 ) for every Ω 1 , Ω 2 ∈ K(V ) and Ω 1 ⊃ Ω 2 .
The space of continuous invariant translation valuations is denoted by Val(V ), the subspace
of Val(V ) of the homogeneous valuations of degree k by Val k (V ) and the subspace of Val(V )
of even valuations (resp. odd valuations) by Val + (V ) (resp. Val − (V )).
Example 2.1.9. The intrinsic volume V i is a continuous invariant translation valuation homogeneous
of degree i.
Remark 2.1.10. The space Val(V ) has structure of infinite dimensional vector space.
The following result of P. McMullen [McM77] gives a decomposition of the space of valuations
depending on the degree, and another depending on the parity
Theorem 2.1.11 ([McM77]). Let n = dim V . Then,
Val(V ) =
n⊕
Val i (V ) and Val(V ) = Val + (V ) ⊕ Val − (V ).
i=0

32 Introduction to valuations
The linear group GL(V ) of invertible linear transformations of V acts transitively on Val(V )
(gφ)(Ω) = φ(g −1 (Ω)) for g ∈ GL(V ), φ ∈ Val(V ), Ω ∈ K(V ).
This action is continuous and preserves the homogeneous degree of the valuation.
Theorem 2.1.12 (Irreducibility Theorem). Let V be an n-dimensional vector space. The
natural representation of GL(V ) at Val + i (V ) and Val− i
(V ) is irreducible for any i ∈ {0, . . . , n}.
That is, there is no proper closed GL(V )-invariant subspace.
From this theorem, it can be proved the following result which relates continuous valuations
with smooth valuations.
Theorem 2.1.13 ([Ale01]). In a vector space V , the smooth translation invariant valuations
are dense in the space of continuous translation invariant valuations.
If V has an Euclidean metric, then every group G subgroup of the orthogonal group, acts on
Val(V ) and it has sense to consider the space Val G (V ) ⊂ Val(V ), i.e. the space of G-invariant
valuations under the action of the group G ⋉ V . The following result by Alesker gives the
necessary and sufficient condition to be this space of finite dimension.
Corollary 2.1.14 ([Ale07a] Proposition 2.6). The space Val G (V ) is finite dimensional if and
only of G acts transitively on the unit sphere of V .
2.2 Hadwiger Theorem
In 1957 Hadwiger proved the following result concerning valuations.
Theorem 2.2.1 ([Had57]). The dimension of the space of continuous translation and O(n)-
invariant valuations is
dim Val O(n) (R n ) = n + 1
and a basis of this space is given by
where V i denotes the i-th intrinsic volume.
V 0 , V 1 , . . . , V n−1 , V n
Remark 2.2.2. From this theorem it follows that in R n the subspace of valuations of homogeneous
degree k ∈ {0, . . . , n} is of dimension 1.
Last remark allows us to prove, in an easy way, some of the classical results of integral
geometry (in R n ), such as reproductive or the kinematic formulas.
Example 2.2.3. • Crofton formula. Let Ω ⊂ R n be a compact convex domain, and let L r
be the space of all planes of dimension r with dL r the (unique up to a constant factor)
invariant density. The measure of the set of planes a convex body in R n can be expressed
in terms of the intrinsic volumes as
∫
χ(Ω ∩ L r )dL r = cV n−r (Ω).
L r
This expression is obtained from the fact that the integral in the left hand side is a
valuation of homogeneous degree (n − r).
At Chapter 4, we study the expression of the measure of complex planes meeting a
domain in CK n (ɛ), and at Chapter 5, the measure of totally real planes of dimension n
meeting a domain.

2.2 Hadwiger Theorem 33
• Reproductive property of the intrinsic volumes. If Ω is a compact convex domain, the
reproductive formula for the intrinsic volumes in R n is given by
∫
V (r)
i
L r
(∂Ω ∩ L r )dL r = cV i (∂Ω)
where L r denotes the space of r-planes in R n and c is a constant depending only on n,
r and i.
Clearly, the integral in the left hand side is a continuous valuation of homogeneous
degree i. By the Hadwiger Theorem the equality follows directly, since the i-th intrinsic
volume is a homogeneous valuation of this degree and the dimension of the space of
homogeneous valuations of degree i is 1. In order to compute the value of c it is used
the so-called template method, which consists on evaluating each side of the equation in
an easy domain (for instance, a sphere) and then compute the value of c. In [San04] it is
given a way to prove this reproductive formula, but using the mean curvature integrals.
The value of c it is also computed.
• Kinematic formula. Although in this work we do not study kinematic formulas, we would
like to give, for completion, the classical kinematic formula of Blaschke-Santaló.
One of the problems of study of the integral geometry consists on measuring the movements
of R n which takes one convex domain to another fixed one. In R n , if we denote
by O(n) the movements group of the space, we obtain the classical kinematic formula
∫
O(n)
χ(Ω 1 ∩ gΩ 2 )dg =
n∑
c n,i V i (Ω 1 )V n−i (Ω 2 ). (2.2)
This formula can be proved from applying twice the Hadwiger Theorem, and then,
applying the obtained formula to spheres of different radius.
Note that the integral on the left hand side is a functional on the first convex domain
Ω 1 , but also on the second convex domain Ω 2 . As a functional on the second convex
domain, from the Hadwiger Theorem, we have
∫
n∑
χ(Ω 1 ∩ gΩ 2 )dg = c i (Ω 1 )V i (Ω 2 )
O(n)
where the coefficients c i (Ω 1 ) depend on Ω 1 . But, the integral under consideration is also
a valuation with respect to Ω 1 , thus, the coefficients c i (Ω 1 ) are valuations and, again by
Hadwiger Theorem, we obtain that it is satisfied
∫
n∑ n∑
χ(Ω 1 ∩ gΩ 2 )dg = c ij V j (Ω 1 )V i (Ω 2 ).
O(n)
i=0
i=0
i=0 j=0
To obtain the expression (2.2), first, note that the desired expression have to be symmetric
with respect to Ω 1 and Ω 2 , hence, c ij = c ji . To prove that most of the constants
c ij vanishes we use the template method, i.e. we apply the equality for a sphere of radius
r and for a sphere of radius R.
Using the invariance with respect to the rotations of a sphere we have
∫
∫ ∫
χ(B r ∩ gB R )dg = χ(B r ∩ (φB R + v))dvdφ (2.3)
O(n)
O(n) R n ∫
= vol(O(n)) χ(B r ∩ (B R + v))dv = vol(O(n))(r + R) n ω n .
R n

34 Introduction to valuations
On the other hand, as the intrinsic volume V i is a homogeneous valuation of degree i we
get
n∑ n∑
n∑
c ij V j (B r )V i (B R ) = c ij r j R i V j (B 1 )V i (B 1 ). (2.4)
i=0 j=0
Thus, in both cases (2.3) and (2.4), we obtained a polynomial on r and R. Comparing
the coefficients we get i + j = n in the last summation in (2.4).
To get the explicit value for the constant, it remains only to use the expression of the
intrinsic volume of sphere of any radius. As the parallel domain at distance r of a sphere
of radius R is a sphere of radius r + R, we can easily compute its intrinsic volume using
the Steiner formula, and we get
2.3 Alesker Theorem
i,j=0
V i (B r ) = r i ( n
i
)
ωn
ω n−i
.
Recently, Alesker gave an analogous theorem of Hadwiger Theorem for the Hermitian standard
space V = C n , with isometry group IU(n) = C n ⋊U(n). As the isometry group of C n is smaller
than the isometry group of R 2n it may happen that some non-invariant valuations under the
isometry group of R 2n is invariant under the isometry group of C n , and this occurs.
Theorem 2.3.1 ([Ale03] Theorem 2.1.1). Let Val U(n) (C n ) be the space of continuous translation
and U(n)-invariant valuations in C n . Then,
( ) n + 2
dim Val U(n) (C n ) = ,
2
and the dimension of the subspace of degree k homogeneous valuations is
min{k, 2n − k}
2
+ 1.
Alesker [Ale03] also gave two bases of continuous isometry invariant valuations on C n . One
of these bases is defined as the integral of the projection volume. That is, let Ω ⊂ C n be a
convex domain and k, l integers such that 0 ≤ k ≤ 2l ≤ 2n, then
∫
C k,l (Ω) :=
G C n,l
V k (Pr Ll (Ω))dL l
where G C n,l denotes the space of complex l-planes in Cn through the origin (see Section 1.5),
Pr Ll (Ω) denotes the orthogonal projection of Ω at L l and V k the k-th intrinsic volume. Valuations
{C k,l } with 0 ≤ k ≤ 2l ≤ 2n define a basis of Val U(n) (C n ). The index k coincides with
the homogeneous degree of the valuation.
The other basis of Val U(n) (C n ) is {U k,p } with k, p integers such that 0 ≤ 2p ≤ k ≤ 2n with
∫
U k,p (Ω) :=
L C n−p
V k−2p (Ω ∩ L n−p )dL n−p (2.5)
where L C n−p denotes the space of complex affine (n − p)-planes of C n (see Section 1.5), and
V k−2p the (k − 2p)-th intrinsic volume. The index k coincides with the homogeneous degree of
the valuation.

2.3 Alesker Theorem 35
Example 2.3.2. We describe explicitily the elements of {U k,p } in C 3 . In C 3 there are ( 5
2)
= 10
linearly independent valuations. For each degree k we have as much valuations as linearly
independent integers p such that
0 ≤ p ≤
min{k, 2n − k}
.
2
Suppose that Ω is a convex domain in C 3 . Then, a basis of Val U(3) (C 3 ) is given by the following
valuations.
k = 0: There is only one value of p, p = 0 and
k = 1: There is only one value of p, p = 0 and
k = 2: There are two values of p, p = 0, 1.
If p = 0 then
U 0,0 (Ω) = V 0 (Ω) = χ(Ω).
U 1,0 (Ω) = V 1 (Ω).
U 2,0 (Ω) = V 2 (Ω).
If p = 1 then
∫
U 2,1 (Ω) = V 0 (Ω ∩ L 2 )dL 2 .
L 2
k = 3: There are two values of p, p = 0, 1.
If p = 0 then
U 3,0 (Ω) = V 3 (Ω).
If p = 1 then
∫
U 3,1 (Ω) = V 1 (Ω ∩ L 2 )dL 2 .
L 2
k = 4: There are two values of p, p = 0, 1.
If p = 0 then
U 4,0 (Ω) = V 4 (Ω).
If p = 1 then
∫
U 4,1 (Ω) = V 2 (Ω ∩ L 2 )dL 2 .
L 2
k = 5: There is only one value of p, p = 0 and
U 5,0 (Ω) = V 5 (Ω) = 1 2 vol(∂Ω).
k = 6: There is only one value of p, p = 0 and
U 6,0 (Ω) = V 6 (Ω) = vol(Ω).
Note that we obtained all intrinsic volumes V j (K), j ∈ {0, . . . , 6}, in the same way as in
R 6 , but at C 3 appear three new linear independent valuations.

36 Introduction to valuations
In the same way as in R n , having a basis for Val U(n) (C n ) allows us to establish a kinematic
formula. The difference is that, now, does not seem possible to find the constants using the
template method. Alesker, in the same paper [Ale03] establish the following result.
Theorem 2.3.3 ([Ale03] Theorem 3.1.1). Let Ω 1 , Ω 2 ⊂ C n be domains with piecewise smooth
boundaries such that for any U ∈ IU(n) the intersection Ω 1 ∩ U(Ω 2 ) has a finite number of
components. Then,
∫
χ(Ω 1 ∩ U(Ω 2 ))dU =
∑ ∑
κ(k 1 , k 2 , p 1 , p 2 )U k1 ,p 1
(Ω 1 )U k2 ,p 2
(Ω 2 ),
U∈IU(n)
p 1 ,p 2
k 1 +k 2 =2n
where the index of the inner summation runs over 0 ≤ p i ≤ k i /2, i = 1, 2, and κ(k 1 , k 2 , p 1 , p 2 )
are constants depending only on n, k 1 , k 2 , p 1 , p 2 .
Theorem 2.3.4 ([Ale03] Theorem 3.1.2). Let Ω ⊂ C n be a domain with piecewise smooth
boundary and 0 < q < n, 0 < 2p < k < 2q. Then,
∫
L C r
U k,p (Ω ∩ L r )dL r =
[k/2]+n−q
∑
p=0
where γ p are constants depending only on n, q, and p.
γ p · U k+2(n−q),p (Ω),
Theorem 2.3.5 ([Ale03] Theorem 3.1.3). Let Ω ⊂ C n be a domain with piecewise smooth
boundary. Then,
∫
[n/2]
∑
χ(Ω ∩ L n )dL n = β p · U n,p (Ω),
L R n
where L R n denotes the space of Lagrangian planes (i.e. totally real planes of dimension n) in
C n and β p are constants depending only on n and p.
The constants for Theorem 2.3.3 are given by Bernig-Fu at [BF08]. These constants were
computed using indirect methods and others bases of valuations in C n . In this work, we give
the constant, in some cases, for Theorems 2.3.4 and 2.3.5.
In [BF08] are given some other bases for Val U(n) (C n ). In Chapters 4 and 5, we use a basis
defined in [BF08] and we extend it to all complex space forms CK n (ɛ). The definition of theses
valuations and its extension in CK n (ɛ) is given at Section 2.4.2.
Finally, we note that Proposition 2.1.14 gives another way to generalize the theory of valuations
in vector spaces. From this proposition, we have that for any group acting transitively
on the sphere can be stated a Hadwiger type theorem, i.e. the space of continuous translation
invariant valuations has finite dimension, thus, it has sense to compute its dimension and give
a basis. An expression for the kinematic formula can also be given.
In this section, we recalled the case in which the acting group is U(n) and befors we studied
SO(n), but there are some known result for other groups.
The groups acting over a sphere are classified and they are (cf. [Bor49], [Bor50], [MS43])
six infinite series
and three exceptional groups
p=0
SO(n), U(n), SU(n), Sp(n), Sp(n) · U(1), Sp(n) · Sp(1)
G 2 , Spin(7), Spin(9).
Alesker and Bernig obtained a Hadwiger type theorem, a kinematic formula (and the
algebraic structure) for some of these groups (cf. [Ale04] for G = SU(2), [Ber08a] for G =
SU(n) and [Ber08b] for G = G 2 and G = Spin(7)).

2.4 Valuations on complex space forms 37
2.4 Valuations on complex space forms
In this section we define some valuations on CK n (ɛ) and we give relations among them.
2.4.1 Smooth valuations on manifolds
The notion of smooth valuation in a differentiable manifold was recently studied (cf. [Ale06a],
[Ale06b], [AF08], [Ale07b], [AB08]). First, a definition of smooth valuation on a manifold was
given, and then it was proved that some important properties, which we do not study in this
work, of smooth valuations, such as the duality property, still hold.
Definition 2.1.7 can be extended for differentiable manifolds.
Definition 2.4.1. Let M be a differentiable manifold and Ω a compact submanifold. Given
a (2n − 1)-form ω in S(M), and a smooth measure η, we consider for any Ω
∫ ∫
η + ω,
Ω
where N(Ω) denotes the normal cycle (cf. [Ale07a]). The obtained functional is called smooth
valuation.
Remark 2.4.2. A more general definition analogue to Definiton 2.1.1 appears in [Ale06b], and
it is called finitely additive measure.
In spaces of constant sectional curvatures, invariant smooth valuations are well-known.
These spaces have the same isotropy group of a point as a point in R n , and from the point of
view of a homogeneous spaces they can be studied in an analogous way. Despite this fact, a
Hadwiger type theorem for continuous valuations (and not only for smooth valuations) it is
not known, i.e. it is not known a basis of continuous translation invariant valuations invariant
also for the isometry group of the space. The dimension of this space of valuations it is not
known an analogous result to Theorem 2.1.13.
Anyway, a big amount of the results in integral geometry are known in these spaces. For
instance, Santaló [San04, page 309] proved that a reproductive formula holds for any real
space form and also obtained an expression for the measure of totally geodesic planes meeting
a convex domain.
In view of these results of Santaló and the knowledge of a basis of continuous invariant
valuations on C n , the aim of this work is to study the classical formulas in integral geometry
described in the last paragraph in complex space forms, i.e. in the standard Hermitian space,
and in the complex projective and hyperbolic space.
2.4.2 Hermitian intrinsic volumes
Bernig and Fu [BF08] defined the Hermitian intrinsic volumes in C n . In this section we recall
this definition and its extension to CK n (ɛ) following [Par02].
Bernig and Fu at [BF08, page 14] defined in T C n , the following invariant 1-forms α, β
and γ and the invariant 2-forms θ 0 , θ 1 , θ 2 and θ s . Let (z 1 , . . . , z n , ζ 1 , . . . , ζ n ) be the canonical
coordinates of T C n ≃ C n × C n with z i = x i + √ −1y i and ζ i = ξ i + √ −1η i . Then,
N(Ω)
θ 0 := ∑ n
i=1 dξ i ∧ dη i , θ 1 := ∑ n
i=1 (dx i ∧ dη i − dy i ∧ dξ i ) ,
θ 2 := ∑ n
i=1 dx i ∧ dy i , θ s := ∑ n
i=1 (dx i ∧ dξ i + dy i ∧ dη i ) ,
α := ∑ n
i=1 ξ idx i + η i dy i , β := ∑ n
i=1 ξ idy i − η i dx i ,
γ := ∑ n
i=1 ξ idη i − η i dξ i .

38 Introduction to valuations
α is the contact form (see Remark 1.3.8) and θ s is the symplectic form of T C n . Recall that a 2-
form ω defined in a manifold of dimension 2m is said symplectic if it is closed, non-degenerated
and ω m ≠ 0.
The previous forms are only defined for ɛ = 0 since canonical coordinates exist only at C n .
But, we can express them in terms of a moving frame on S(C n ) (and, then extend them for
any ɛ ∈ R). The expression in terms of a moving frame allows us to prove that these forms are
well-defined, i.e. they do not depend on the chosen coordinates. Let (x, v) ∈ S(CK n (ɛ)) and
let {x; e 1 := v, Je 1 , . . . , e n , Je n } be a J-moving frame defined in a neighborhood of x. Then
θ 0 =
θ 1 =
θ 2 =
n∑
α 1i ∧ β 1i ,
i=2
n∑
(α i ∧ β 1i − β i ∧ α 1i ), (2.6)
i=2
n∑
α i ∧ β i ,
i=2
where α i , β i , α ij , β ij are the forms given at (1.12) but interpreted as forms in S(CK n (ɛ)).
Remark 2.4.3. From the expression of θ 0 , θ 1 and θ 2 in terms of a moving frame we can define
these 2-forms in S(CK n (ɛ)) for any ɛ.
Remark 2.4.4. In [Par02] invariant 2-forms at CK n (ɛ) are defined in the same way as before
(see Section 2.4.3).
Proposition 2.4.5 ([Par02] Proposition 2.2.1). The algebra Ω ∗ (S(CK n (ɛ))) of R-valued invariant
differential forms on the unit tangent bundle S(CK n (ɛ)) is generated by
α, β, γ, θ 0 , θ 1 , θ 2 , θ s .
From this proposition, Bernig and Fu define the families of (2n−1)-forms {β k,q } and {γ k,q }
at S(C n ), but from the definition of θ 0 , θ 1 and θ 2 at S(CK n (ɛ)) these families of forms can be
defined in the same way at S(CK n (ɛ)). By the previous proposition we have, as it is note in
[Par02], that all (2n − 1)-form invariant on S(CK n (ɛ)) such that they do not vanish over the
normal bundle of a domain Ω (cf. Lemma 1.3.12) are the ones given in the next definition.
Definition 2.4.6. Let k, q ∈ N be such that max{0, k − n} ≤ q ≤ k 2
differential (2n − 1)-forms at S(CK n (ɛ)) are defined
< n. Then, the following
where
β k,q := c n,k,q β ∧ θ n−k+q
0 ∧ θ k−2q−1
1 ∧ θ q 2 , k ≠ 2q
γ k,q := c n,k,q
2 γ ∧ θn−k+q−1 0 ∧ θ k−2q
1 ∧ θ q 2 , n ≠ k − q
1
c n,k,q :=
q!(n − k + q)!(k − 2q)!ω 2n−k
and ω 2n−k denotes the volume of the Euclidean ball of radius 1 and dimension 2n − k.
Definition 2.4.7. Given a regular domain Ω ⊂ CK n (ɛ), forms β k,q and γ k,q define the following
invariant valuations (see Section 2.4.3) in CK n (ɛ) (for max{0, k − n} ≤ q ≤ k 2 < n)
∫
∫
B k,q (Ω) := β k,q (k ≠ 2q) and Γ k,q (Ω) = γ k,q (n ≠ k − q)
N(Ω)
N(Ω)
where N(Ω) denotes the normal fiber bundle of Ω (see Definition 1.3.10).

2.4 Valuations on complex space forms 39
In C n the previous valuations satisfy B k,q (Ω) = Γ k,q (Ω) since dβ k,q = dγ k,q . For ɛ ≠ 0 no
form β k,q has the same differential as γ k,q .
The differential of the forms θ 0 , θ 1 and θ 2 is given in [BF08] for ɛ = 0. From the structure
equations in CK n (ɛ) (cf. [KN69]), in the same way as in [Par02], we compute the differential
for any ɛ.
Lemma 2.4.8. In S(CK n (ɛ)) it is satisfied
dα = −θ s , dθ 0 = −ɛ(α ∧ θ 1 + βθ s ),
dβ = θ 1 , dθ 1 = dθ 2 = dθ s = 0
dγ = 2θ 0 − 2ɛ(α ∧ β + θ 2 ).
In the following proposition we give the relation between valuations {B k,q (Ω)} and {Γ k,q (Ω)}
in CK n (ɛ).
Proposition 2.4.9. In CK n (ɛ), for any pair of integers (k, q) such that max{0, k − n} < q <
k/2 < n it is satisfied
c n,k,q
Γ k,q (Ω) = B k,q (Ω) − ɛ B k+2,q+1 (Ω)
c n,k+2,q+1
(q + 1)(2n − k)
= B k,q (Ω) − ɛ
2π(n − k + q) B k+2,q+1(Ω).
Proof. Denote by I the ideal generated by α, dα and all the exact forms in N(Ω) (see Definition
1.3.10). If two forms λ and ρ of degree 2n − 1 coincide modulo I, then by Lemma 1.3.12
∫ ∫
λ = ρ.
Thus, it is enough to prove
N(Ω)
c n,k,q
N(Ω)
γ k,q ≡ β k,q − ɛ β k+2,q+1 mod I. (2.7)
c n,k+2,q+1
Consider the form η = (θ s − β ∧ γ) ∧ θ n−k+q−1
0 θ k−2q−1
1 θ q 2 . As dη is exact we have dη ≡ 0
mod I. On the other hand, from the differentials given in Lemma 2.4.8 we obtain
dη ≡ −γθ n−k+q−1
0 θ k−2q
1 θ q 2 + 2βθn−k+q 0 θ k−2q−1
1 θ q 2 − 2ɛβθn−k+q−1 0 θ k−2q−1
1 θ q+1
2 mod I.
Using the definition of γ k,q and β k,q we get the relation (2.7).
Remark 2.4.10. For n = 2, 3, the previous relations are given in [Par02].
By the relation in Proposition 2.4.9, we define the Hermitian intrinsic volumes in CK n (ɛ).
Definition 2.4.11. For max{0, k−n} ≤ q ≤ k 2
< n, we define the Hermitian intrinsic volumes
µ k,q in CK n (ɛ)
{
Bk,q (Ω) si k ≠ 2q
µ k,q (Ω) :=
(2.8)
Γ 2q,q (Ω) si k = 2q.
Remark 2.4.12. In C n , Hermitian intrinsic volumes form a basis of continuous valuations
invariant under the isometry group of C n (cf. [BF08]).
In the previous definition of µ k,q we take an arbitrary choice, despite of the relations in
Proposition 2.4.9. It would be interesting to know if there is a better choice, such that it
satisfies some more natural geometric or algebraic properties.

40 Introduction to valuations
2.4.3 Relation between Hermitian intrinsic volumes and the valuations given
by Park
We give, by completeness, the valuations defined by Park in CK n (ɛ).
Definition 2.4.13. Denote θ 00 = −α ∧ β + θ 2 , θ 01 = −β ∧ γ + θ s , θ 10 = −α ∧ γ + θ 1 , θ 11 = θ 0 .
Let κ = σ ∧ {a, b, c} where σ ∈ {α, β} and {a, b, c} = 1
a!b!c! θa 11 ∧ θb 00 ∧ θc 10 . Then
∫
Φ κ (Ω) =
N(Ω)
are smooth valuation invariant under the action of the isometry group of CK n (ɛ).
The relation between these valuations and the Hermitian intrinsic volumes is given straightforward
from the definition of each valuation.
Proposition 2.4.14. Let Ω ⊂ CK n (ɛ) be a regular domain. Then,
1
B k,q (Ω) =
Φ β{n−k+q,q,k−2q−1} (Ω),
(k − 2q)ω 2n−k
1
Γ k,q (Ω) =
Φ γ{n−k+q−1,q,k−2q} (Ω).
2(n − k + q)ω 2n−k
κ
2.4.4 Other curvature integrals
Let M be a Kähler manifold of complex dimension n and suppose that S ⊂ M is a real
hypersurface. Then, we can canonically define a distribution of complex dimension n − 1 in
the tangent fiber bundle of S in the following way.
Let N x be the normal fiber bundle of S at x. Let J be the complex structure in M. The
vector JN x is a tangent vector to S at x. Consider the orthogonal vectors to JN x inside the
tangent space of S at x. These form a complex subspace of dimension n − 1. Denote by D
the distribution defined by these subspaces. Then, we define the mean curvature integrals
restricted to the distribution D as
Definition 2.4.15. Let S be a hypersurface of a Kähler manifold M of complex dimension n.
If x ∈ S, we denote the second fundamental form of S at x by II x and the second fundamental
form restricted to D by II x | D . The r-th mean curvature integral of S restricted at D, 1 ≤ r ≤
2n − 2, is defined as
( ) 2n − 2 −1 ∫
Mr D (S) =
σ r (II x | D )dx
r
where σ r (II x | D ) denotes the r-th symmetric elementary function of II x | D .
Along this work, we use the idea of restricting mean curvature integrals to the distribution
D. The first mean curvature integral restricted to D will play an important role, i.e. the
integral of the trace of the second fundamental form restricted to the distribution. Also the
integral over the normal curvature JN will have an important role. If Ω is a regular domain,
we have the following relations
∫
(2n − 1)M 1 (∂Ω) − (2n − 2)M1 D (∂Ω) =
∂Ω
S
k n (JN)dp = 2ω 2 Γ 2n−2,n−1 (Ω). (2.9)

2.4 Valuations on complex space forms 41
2.4.5 Relation between the Hermitian intrinsic volumes and the second
fundamental form
In this section we give another expression for the Hermitian intrinsic volumes in terms of the
second fundamental form.
In the proof of Theorem 4.3.1 we shall use some properties, interesting by their own, of
the expression of the invariant (2n − 1)-forms expressed in terms of the second fundamental
form of ∂Ω, and not only in terms of the connection forms of a moving frame. In order to give
this expression in terms of the second fundamental form (with respect to a fixed basis) it is
necessary to consider the pull-back of the following canonical map
ϕ : ∂Ω −→ N(Ω)
x ↦→ (x, N x ) . (2.10)
Let us study some properties of ϕ ∗ (β k,q ) and ϕ ∗ (γ k,q ).
Let x ∈ ∂Ω ⊂ CK n (ɛ) and let {e 1 = ϕ(x), e 1 = Je 1 , . . . , e n , e n = Je n } be a J-moving
frame defined in a neighborhood of x. Consider the 1-forms {α i , β i , α 1j , β 1j }, and the 2-forms
{θ 0 , θ 1 , θ 2 , θ s } given at (2.6).
Notation 2.4.16. In order to simplify the notation in the following expressions we denote β i
by α i
and β 1i by α 1i
and we define I := {1, 2, 2, . . . , n, n}.
Now, using the relation between the connection forms α 1i , i ∈ I, and the second fundamental
form
α 1i = ∑ h ij α i , (2.11)
j∈I
we obtain
Lemma 2.4.17. In the previous notation,
ϕ ∗ (β) = α 1 ,
ϕ ∗ (γ) = ∑ h 1j α j ,
j∈I
n∑ ∑
ϕ ∗ (θ 0 ) = h ij h il
α j ∧ α l ,
ϕ ∗ (θ 1 ) =
ϕ ∗ (θ 2 ) =
i=2 j,l∈I
⎛
n∑
⎝ ∑ h ij
α i ∧ α j − ∑
j∈I
l∈I
i=2
n∑
α i ∧ α i
.
i=2
h il α i
∧ α l
⎞
⎠ ,
On the other hand, each form ϕ ∗ (β k,q ) is a form of maximum degree, and, thus, a multiple
of the volume element dx = α 1 ∧ α 2 ∧ α 2 ∧ · · · ∧ α n of ∂Ω. Thus, ϕ ∗ (β k,q ) is determined by this
multiple, which can be interpreted as a polynomial with variables the entries of the second
fundamental form h ij (with respect to the fixed J-moving frame).
In [Par02], it is computed explicitly the pull-back of the forms β k,q and γ k,q for dimensions
n = 2, 3. In the following lemma we give some general properties for the pull-back of these
forms for any dimension n.
Lemma 2.4.18. Let Ω ⊂ CK n (ɛ) be a regular domain and let ϕ : ∂Ω → N(Ω) be the canonical
map defined at (2.10). Let us fix a point x ∈ ∂Ω and a J-moving frame {e 1 = ϕ(x), e 1 =
Je 1 , . . . , e n , e n = Je n } at x. If
ϕ ∗ (β k,q ) = Q k,q dx,
ϕ ∗ (γ k,q ) = P k,q dx

42 Introduction to valuations
where dx is the volume element of ∂Ω, then
1. Q k,q is a polynomial of degree 2n − k − 1 with variables the entries of the second fundamental
form h ij = II(e i , e j ), i, j ∈ {2, 2, . . . , n};
2. P k,q is a polynomial of degree 2n − k − 1 with variables the entries of the second fundamental
form h ij = II(e i , e j ), i, j ∈ {1, 2, 2, . . . , n};
3. each monomial of P k,q containing only entries of the form h ii also contains h 11 and
exactly n + q − k − 1 factors of the form h jj h jj
, i ∈ {1, 2, 2, . . . , n}, j ∈ {2, . . . , n};
4. the polynomials P k,q and Q k,q can be written as a sum of minors of the second fundamental
form with rank r = 2n − k − 1;
5. among the minors described at 4. appear all minors centered at the diagonal with degree r
containing h 11 for P k,q , and not containing h 11 for Q k,q . It also appears all non-centered
minors such that the indices of the rows and the indices of the columns determining a
minor satisfy
(a) contain n − k + q indices, in the case of Q k,q , and n − k + q − 1, in the case of P k,q ,
such that, if the index j appears as an index in the rows (resp. columns), then the
index j also appears as an index in the rows (resp. columns) of the minor. We say
that the index j is paired in the rows (or in the columns);
(b) contain k − 2q − 1 indices non-paired neither in the rows nor in the columns for
Q k,q , and k − 2q for P k,q ;
(c) if the index j is in the non-paired indices of the rows, then the index j is not in the
index of the columns.
Proof. From Lemma 2.4.17 we have
and
⎛
⎞
n∑ ∑
ϕ ∗ (β k,q ) = c n,k,q α 1 ∧ ⎝ h ij h il
α j ∧ α l
⎠
ϕ ∗ (γ k,q ) = c n,k,q
2
i=2 j,l∈I
n+q−k
⎛ ⎛
⎞⎞
n∑
∧ ⎝ ⎝ ∑ h ij
α i ∧ α j − ∑ h il α i
∧ α l
⎠⎠
i=2 j∈I
l∈I
∧ (2.12)
k−2q−1
⎛ ⎞ ⎛
⎞
⎝ ∑ n+q−k−1
n∑ ∑
h 1j α j
⎠ ∧ ⎝ h ij h il
α j ∧ α l
⎠ ∧
j∈I
i=2 j,l∈I
⎛ ⎛
⎞⎞
n∑
∧ ⎝ ⎝ ∑ h ij
α i ∧ α j − ∑ h il α i
∧ α l
⎠⎠
i=2 j∈I
l∈I
k−2q
( n∑
) q
∧ α i ∧ α i
,
i=2
( n∑
) q
∧ α i ∧ α i
.
i=2
Thus, as ϕ ∗ (β k,q ) and ϕ ∗ (γ k,q ) are differential forms defined on ∂Ω of maximum degree, we
have that ϕ ∗ (β k,q ) = Q k,q dx and ϕ ∗ (γ k,q ) = P k,q dx satisfy that Q k,p and P k,q are polynomials
of degree 2n − k − 1. Moreover, polynomials P k,q cannot contain h 11 since this variable is
multiplied by α 1 (see formula (2.11)), but this differential form is a common factor in the
expression ϕ ∗ (β k,q ).

2.4 Valuations on complex space forms 43
In order to prove 3. we observe that terms in the expression ϕ ∗ (γ k,q ) containing only entries
of the type h ii are
( n∑
) n+q−k−1 ( n∑
) k−2q ( n∑
) q
c n,k,q
2 h 11 α 1 ∧ h ii h ii
α i ∧ α i
∧ h ii
α i ∧ α i
− h ii α i
∧ α i ∧ α i ∧ α i
.
i=2
i=2
i=2
So, the variable h 11 always appears and there are exactly n + q − k − 1 factors of the form
h jj h jj
, which come from ( ∑ n
i=2 h iih ii
α i ∧ α i
) n+q−k−1 .
A minor of rank r of a matrix is defined choosing r rows and r columns of the matrix and
then taking the determinant of the square submatrix. To prove 4. and 5. we study in more
detail the expression (2.12). (An analogous study can be done in the case ϕ ∗ (γ k,q ).) First, note
that factors α 1 and ϕ ∗ (θ 2 ) do not give any term of the second fundamental form and, thus,
they do not influence in the minor (but they do influence in which minors can be constructed).
Thus, we have to prove that the expression
ϕ ∗ (θ n+q−k
0 ) ∧ ϕ ∗ (θ k−2q−1
1 ), (2.13)
is a form of degree 2n−2q −2 where each term α i1 ∧α i2 ∧· · ·∧α i2n−2q−2 goes with a summation
of minors.
Developing (2.13) we have that it is equivalent to
⎛
⎞
n∑
n∑
ϕ ∗ ⎝( α 1i ∧ α 1i
) n+q−k ∧ ( (α j ∧ α 1j
− α j ∧ α 1j )) k−2q−1 ⎠ .
i=2
j=2
To develop this expression, first, we chose a := n + q − k values {i 1 , . . . , i a } for the index i
of the first summation and b := k − 2q − 1 values {j 1 , . . . , j b } (with i k , j l ∈ {2, . . . , n}) for
the index j of the second summation. Note that some indexes can be repeated. Thus, we get
( n−1
n+q−k
) (
·
n−1
k−2q−1)
summands
ϕ ∗ (α 1i1 ∧ α 1i1
∧ · · · ∧ α 1ia ∧ α 1ia
∧ (α j1 ∧ α 1j1
− α j1
∧ α 1j1 ) ∧ · · · ∧ (α jb ∧ α 1jb
− α jb
∧ α 1jb )),
which can be decomposed, for example, in the form
ϕ ∗ (α 1i1 ∧ α 1i1
∧ · · · ∧ α 1ia ∧ α 1ia
∧ α j1 ∧ α 1j1
∧ · · · ∧ α jb ∧ α 1jb
)
= α j1 ∧ · · · ∧ α jb ϕ ∗ (α 1i1 ∧ α 1i1
∧ · · · ∧ α 1ia ∧ α 1ia
∧ α 1j1
∧ · · · ∧ α 1jb
).
From (2.11) the form in which we take pull-back can be expressed as the summation of the
minors with rows given by the indices I := {i 1 , i 1 , . . . , i a , i a , j 1 , . . . , j b }, and by columns each
of the possible permutations of the elements without repetition among the indexes in J :=
I \ {1, j 1 , . . . , j b }. Note that index α 1 cannot be taken since we are considering the form in
(2.12), which is multiplied by α 1 . (In the case of ϕ ∗ (γ k,q ) we do not have this restriction, and,
so, can also appear minors with the term h 11 .)
If we chose for J the same indexes as in I, then we get all the minors centered at the
diagonal. Condition 5.(c) is obtained directly from the fact that the indices which determines
the columns have to be in J , and if j k is an index in the rows, the index j k is not in J .
Conditions 5.(a) and 5.(b) are obtained when we recall that we are not studying the differential
form in (2.13) but in (2.12), i.e. we have to take the product with α 1 ∧ ( ∑ n
i=2 α i ∧ α i
) q .
But, if this product contains the form α k , then it also contains the form α k
(except for k = 1).
Thus, to complete the form in (2.13) to a (2n − 1)-differential form we have to take the products
α k ∧ α k
, so that, in order to not obtain a vanishing term, the quantity of paired indices
in the rows and in the columns must be the same, and, thus, it also coincides the quantity of
no-paired indices.

44 Introduction to valuations
Remarks 2.4.19. 1. Each of the minor goes with a constant depending on the number of
permutations that allows us to obtain it. Thus, not all the minors have the same constant,
but, for instance all the minors (fixed β k,q or γ k,q ) centered at the diagonal have the same
one.
2. The degree of the polynomial given by β k,q or γ k,q does not depend on q, just on k, but
two polynomials coming from β k,q and β k,q ′ (or γ k,q and γ k,q ′) are distinguished by the
number of paired indexes.
Example 2.4.20. We give the explicit relation between some Hermitian intrinsic volumes and
the second fundamental form.
1. ϕ ∗ (γ 00 ) = c n,0,0
2 ϕ∗ (γ ∧ θ0 n−1 ) = (n − 1)! c n,0,0
det(II)dx = 1 det(II)dx.
2
2nω 2n
2. ϕ ∗ (β 10 ) = c n,1,0 ϕ ∗ (β ∧ θ n−1
0 ) = (n − 1)!c n,1,0 det(II| D )dx = 1
ω 2n−1
det(II| D )dx.
3. ϕ ∗ (γ 2n−2,n−1 ) = c n,2n−2,n−1
ϕ ∗ (γ∧θ n−1
2
4. ϕ ∗ (β 2n−2,n−2 ) = c n,2n−2,n−2 ϕ ∗ (β ∧ θ 1 ∧ θ n−2
2 ) =
2 ) = c n,2n−1,n−1(n − 1)!
2
k n (JN)dx = k n (JN) dx
2ω 2
.
(n − 2)!
(n − 2)!2ω 2
tr(II| D )dx = tr(II| D ) dx
2ω 2
.
5. ϕ ∗ (β 2n−1,n−1 ) = c n,2n−1,n−1 ϕ ∗ (β ∧ θ n−1
2 ) = c n,2n−1,n−1 (n − 1)!α 1 ∧ α 1 ∧ · · · ∧ α n = dx 2 .

Chapter 3
Average of the mean curvature
integral
For the real space forms (R n , S n and H n ), it is known that the reproductive property holds for
mean curvature integrals. That is, given a regular domain Ω, it is satisfied (cf. Example 2.2.3)
∫
M r
(s)
L s
(∂Ω ∩ L s )dL s = cM r (∂Ω).
On the other hand, by Section 2.3, this property may not hold in C n , when we integrate over
the space of complex planes. Thus, it is natural to study, in C n , the value of
∫
L C s
M (s)
r (∂Ω ∩ L s )dL s .
In the same way, we will study the value of this integral but in the other complex space
forms, CP n and CH n . Recall that we denote by CK n (ɛ) the space of constant holomorphic
curvature 4ɛ.
In this chapter we deduce the expression of the integral of M (s)
1 (∂Ω ∩ L s) in terms of the
mean curvature integral of the convex domain M 1 (∂Ω) and the integral of the normal curvature
in the direction JN, ∫ ∂Ω k n(JN) (see Theorem 3.3.2). We also find a partial result for the
integral over any other mean curvature integral M r
(s) (∂Ω∩L s ), 0 ≤ r ≤ 2s−1 (see Proposition
3.2.2).
3.1 Previous lemmas
First of all, we state some lemmas that will be necessary in order to prove Theorem 3.3.2 and
some other results.
Lemma 3.1.1. Let V be a complex vector space of complex dimension 2 endowed with an inner
product 〈 , 〉 compatible with the complex structure J and let {e 1 , e 2 , e 3 , e 4 } be an orthonormal
basis of V . Then 〈e a , Je b 〉 2 = 〈e c , Je d 〉 2 with {a, b, c, d} = {1, 2, 3, 4}.
Proof. We express Je b and Je d in terms of the orthonormal, and we obtain
Je b = 〈Je b , e a 〉e a + 〈Je b , e c 〉e c + 〈Je b , e d 〉e d = Ae a + Be c + Ce d ,
Je d = 〈Je d , e a 〉e a + 〈Je d , e b 〉e b + 〈Je d , e c 〉e c = De a + Ee b + F e c .
45

46 Average of the mean curvature integral
Now, using that 〈Je b , Je b 〉 = 〈Je d , Je d 〉 = 1, 〈Je b , Je d 〉 = 0 and 〈Je b , e d 〉 = −〈Je d , e b 〉, we
get
and
A 2 + B 2 + C 2 = 1,
D 2 + E 2 + F 2 = 1,
AD + BF = 0,
C = −E,
A 2 + B 2 = D 2 + F 2 and A 2 D 2 = B 2 F 2 .
Finally, we substitute D 2 = A 2 + B 2 − F 2 in the second equation and we obtain A 2 = F 2 .
Lemma 3.1.2. If u ∈ S 2n−3 , then
∫
S 2n−3 〈u, v〉dv = 0
and
∫
S 2n−3 〈u, v〉 2 dv = ω 2n−2 .
Proof. The first equality follows since the integral is over an odd function. For the second one,
we decompose v = cos θu + sin θw with w ∈ 〈u〉 ⊥ , then using polar coordinates with respect
to u, we have
∫S 2n−3 〈u, v〉 2 dv = O 2n−4
∫ π
0
cos 2 θ sin 2n−4 θdθ = O 2n−4
O 2n−4+2+1
O 2 O 2n−4
= ω 2n−2 .
where O n denotes the volume of the n-dimensional Euclidean sphere and ω n the volume of the
the n-dimensional Euclidean ball.
The following lemma gives a generalized version of the Meusiner Theorem.
Lemma 3.1.3. Let S ⊂ M be a hypersurface of class C 2 of a Riemannian manifold M, p ∈ S
and L ⊂ T p M a vector subspace. We denote by II S the second fundamental form of S and
by II L C the second fundamental form of C = S ∩ exp p L as a hypersurface of exp p L. We also
denote u = T p S ∩ L. Then,
σ i (II S | u ) = cos i θσ i (II| L C)
where θ denotes the angle at p between a normal vector of S and a normal vector of C in
exp p L, and σ i (Q) denotes the i-th symmetric elementary function of the bilineal form Q.
Proof. If A ⊂ B ⊂ M are submanifolds, then we denote the second fundamental form of A as
a submanifold of B by h B A : T pA × T p A → (T p A) ⊥ . If B = M, we just put h A instead of h M A .
Then, for all X, Y ∈ T p C
h C (X, Y ) = h L C(X, Y ) + h L (X, Y ) = h L C(X, Y )
since the second fundamental form of L vanishes at p. Moreover,
h C (X, Y ) = h S C(X, Y ) + h S (X, Y ).
Let N be a normal vector to S. Note that h S C
(X, Y ) is a multiple of a normal vector to C in
S, so 〈h S C (X, Y ), N〉 = 0 (for X, Y ∈ T pC).
If X, Y ∈ T p C, then
II S (X, Y ) := 〈h S (X, Y ), N〉 = 〈h C (X, Y ) − h S C(X, Y ), N〉
= 〈h C (X, Y ), N〉 = 〈h L C(X, Y ), N〉 = 〈II L C(X, Y )n, N〉

3.1 Previous lemmas 47
where n denotes a normal vector of C in L. So,
II L C(X, Y ) = 1
〈N, n〉 II S(X, Y ). (3.1)
Since σ i (II L C ) is the sum of the minors of order i of IIL C , by replacing by (3.1) each entry of the
second fundamental form, we obtain the result.
The following lemma generalizes in C n a result given by Langevin and Shifrin [LS82] in
R n .
Lemma 3.1.4. Let E be a complex vector space of complex dimension n and let II be a real
bilinear form defined on E. We denote by G C n,s the Grassmanian of s-dimensional complex
planes on E. Then,
∫
tr(II| V )dV = s vol(GC n,s)
tr(II| E ).
n
G C n,s
Proof. First, recall that
is a fibration for each s ∈ {1, . . . , n − 1}.
U(n − s) × U(s) −→ U(n) −→ G C n,s (3.2)
We prove the case dim C V ≤ n 2
by induction on the complex dimension of V . The case
dim C V > n 2
can be proved using similar arguments.
Suppose dim C V = 1, that is, s = 1. Then,
∫
G C n,1
tr(II| V )dV =
∫
1
tr(II|
vol(U(n − 1))vol(U(1))
V 1
1
)dU
U(n)
since tr(II| V 1
1
) is constant along the fiber. We denote by V1 1 the complex vector subspace
generated by the first column of the matrix U ∈ U(n). In general, for U ∈ U(n), we will
denote by Va b the complex vector subspace generated by the columns b to b + a − 1. The
subscript a denotes the dimension of Va b , or equivalently, the number of columns we consider
and the upperscript b denotes from which column we start to consider them. Then
∫
U(n)
tr(II| V 1
1
)dU = 1 n
= 1 n
∫
∫
U(n)
U(n)
(tr(II| V 1
1
) + tr(II| V 2
1
) + · · · + tr(II| V n
1
))dU
tr(II| E )dU = vol(U(n)) tr(II| E ).
n
Thus,
∫
G C n,1
tr(II| V )dV =
vol(U(n))
nvol(U(n − 1))vol(U(1)) tr(II| E) = vol(GC n,1 ) tr(II| E ).
n
Suppose now that the result is true till dim C V = r −1. We shall prove it for dim C V = r ≤
n
2 . If R denotes the remainder of n r
, then R < r and we can apply the induction hypothesis in

48 Average of the mean curvature integral
R at equality (*). Thus, using similar arguments as before, we obtain
∫
G C n,r
tr(II| V ) =
∫
1
tr(II|
vol(U(n − r))vol(U(r))
V 1 r
)
U(n)
1
(tr(II| V 1 r
) + tr(II| V 2 r
) + · · · + tr(II| V
n−R−r+1
=
vol(U(n − r))vol(U(r))⌊ n r
∫U(n) ⌋
( ∫
1
=
vol(U(n − r))vol(U(r))⌊ n r ⌋ U(n)
(∗)
=
∫
tr(II| E ) −
U(n)
tr(II| V
n−R+1)
R
vol(U(n))tr(II| E ) − vol(U(n − R))vol(U(R)) ∫ G
tr(II| C VR )
n,R
=
vol(U(n − r))vol(U(r))⌊ n r
(
⌋ )
vol(U(n)) − vol(U(n − R))vol(U(R))vol(G C n,R ) R n
tr(II| E )
= vol(G C r
n,r)
n − R
= vol(G C n,r) r n tr(II| E)
and the result follows when 2s ≤ n.
vol(U(n − r))vol(U(r))⌊ n r
(
⌋
1 − R )
tr(II| E )
n
)
r
))
3.2 Integral of the r-th mean curvature integral over the space
of complex s-planes
Along this chapter we follow some conventions which we state in the following paragraphs.
We denote by S ⊂ CK n (ɛ) a hypersurface of class C 2 , compact and oriented (possibly with
boundary). Given a complex s-plane L s intersecting S, it is said that L s is in generic position
if S ∩ L s is a submanifold of dimension 2s − 1 in CK n (ɛ). For hypersurfaces of class C 2 , the
subset of generic planes (intersecting S) has full measure. Thus, we suppose that each complex
s-plane is in generic position. Note that S ∩ L s (if L s is a complex s-plane in generic position)
is a hypersurface in L s
∼ = CK s (ɛ).
Suppose that N denotes a unit normal vector field on S. In this chapter we take, in S ∩ L s
as a submanifold in S, the normal vector field Ñ such that the angle between N and Ñ is
acute. Along the proofs in this chapter, we denote e s := ±JÑ.
Note that if p ∈ S ∩ L s and Ñp is the chosen normal vector field in S ∩ L s inside L s then
JÑ ∈ T p(S ∩ L s ). Indeed, L s
∼ = CK s (ɛ), thus, the same structure for hypersurfaces hols inside
L s .
We denote by E ⊂ T p CK n (ɛ), p ∈ S ∩ L s , the orthogonal subspace to the space generated
by {N, JN, Ñ, JÑ}. Note that exp p(E) ∼ = CK n−2 (ɛ) and it is univocally determined for each
L s .
The fact stated in the following remark is used implicitly along this chapter, specially to
define the moving frames g and g ′ in the proof of the next proposition.
Remark 3.2.1. Let V n be an n-dimensional Hermitian space with complex structure J, H ⊂ V n
a real hyperplane and W s ⊂ V n a complex subspace of dimension s.
Consider the subspace H ∩ W s and denote by N ′ an orthonormal vector to H ∩ W s in W s ,
and D ′ = 〈N ′ , JN ′ 〉 ⊥ ∩ W s .
Consider also the subspaces H ∩ Ws
⊥
in Ws ⊥ , and D ′′ = 〈N ′′ , JN ′′ 〉 ⊥ ∩ W s ′′ .
and denote by N ′′ an orthonormal vector to H ∩ W ⊥ s

3.2 Integral of the r-th mean curvature integral 49
Denote by N an orthonormal vector to H in V , and D = 〈N, JN〉 ⊥ . Then,
V n = D⊥〈JN〉 R ⊥〈N〉 R
= D ′ ⊥〈JN ′ 〉 R ⊥〈N ′ 〉 R ⊥D ′′ ⊥〈JN ′′ 〉 R ⊥〈N ′′ 〉 R .
Indeed, from section 2.4.4 given a real hyperplane W in a Hermitian space V there exists
a canonical decomposition of V as
V = D⊥〈JN〉 R ⊥〈N〉 R ,
where N is an orthogonal vector to W in V . Applying this fact to V = W s and to V = W ⊥ s
we get the result.
The following proposition shall be essential to prove Theorem 3.3.2 and other results,
since it gives a first expression of the integral over the space of complex s-planes of the mean
curvature integral in terms of an integral on the boundary of the domain.
Proposition 3.2.2. Let S ⊂ CK n (ɛ) be a compact (possibly with boundary) hypersurface of
class C 2 oriented by a normal vector N, and let r, s ∈ N such that 1 ≤ s ≤ n and 0 ≤ r ≤ 2s−1.
Then
∫
( ) 2s − 1 −1 ∫ ∫ ( ∫ )
M r (s)
|〈JN, e s 〉|
(S∩L s )dL s =
L C r
s
S( 2s−r
RP 2n−2 G C (1 − 〈JN, e
n−2,s−1
s 〉 2 ) s−1 σ r(p; e s ⊕ V )dV
)de s dp,
where e s ∈ T p S unit vector, V denotes a complex (s−1)-plane by p contained in {N, JN, e s , Je s } ⊥ ,
σ r (p; e s ⊕ V ) denotes the r-th symmetric elementary function of the second fundamental form
of S restricted to the real subspace e s ⊕V and the integration over RP 2n−2 denotes the projective
space of the unit tangent space of the hypersurface.
Remark 3.2.3. Using the previous remarks, it follows that the product |〈JN, e s 〉| in the last
proposition gives the cosine of the acute angle between the normal vector to the hypersurface
S in CK n (ɛ) and a normal vector to S ∩ L s in L s , that is,
|〈JN, e s 〉| = |〈N, Ñ〉|.
Proof. Let L s be a complex s-plane such that S ∩ L s ≠ ∅ and let p ∈ S ∩ L s . We denote
by ˜σ r the r-th symmetric elementary function of the second fundamental form of S ∩ L s as a
hypersurface of L s . Then, by definition
∫
( ) 2s − 1 −1 ∫ ∫
M r (s) (S ∩ L s )dL s =
˜σ r (s)dx dL s .
r S∩L s≠∅ S∩L s
L C s
We shall prove the result using moving frames adapted to S ∩ L s , L s or S.
Let g = {e 1 , e 1 = Je 1 , e 2 , e 2 = Je 2 , . . . , e s , w s , e s+1 , e s+1 = Je s+1 . . . , e n , N} be a moving
frame adapted to S ∩ L s and S (cf. Remark 3.2.1). That is, {e 1 , e 1 , . . . , e s } is an orthonormal
basis of T p (S ∩L s ), {e s+1 , e s+1 , . . . , e n } is an orthonormal basis of T p S ∩(T p L s ) ⊥ , N is a normal
vector field to T S and w s completes to an orthonormal basis of T p CK n (ɛ). We denote by
{ω 1 , ω 1 , . . . , ω s−1 , ω s−1 , ω s , ω˜s , ω s+1 , ω s+1 , . . . , ω n , ωñ}
the dual basis of the vectors in g and by {ω ij }, i, j ∈ {1, 1, . . . , s, ˜s, s + 1, s + 1, . . . , n, ñ}, the
connection forms (cf. (1.15)).
Let g ′ = {e ′ 1 = e 1, e ′ 1 = Je 1, e ′ 2 = e 2, e ′ 2 = Je 2, ..., e ′ s = e s , e ′ s = Je s, e ′ s+1 = e s+1, e ′ s+1 =
Je s+1 , ..., e ′ n = e n , e ′ n = Je n} be a moving frame adapted to S ∩ L s and L s . That is,

50 Average of the mean curvature integral
{e ′ 1 , e′ 1 , ..., e′ s} is an orthonormal basis of T p (L s ∩ S) and {e ′ 1 , e′ 1 , . . . , e′ s, e s ′ } is an orthonormal
basis of T p L s . Denote by
{ω 1, ′ ω ′ 1 , . . . , ω′ n, ω n}
′
the dual basis of vectors in g ′ and by {ω ij ′ } the connection forms, i, j ∈ {1, 1, . . . , n, n}.
As the base g and g ′ are constituted by orthonormal vectors, we can easily give the relation
among the elements in the frame g ′ and the ones in g just expressing the vectors in g ′ in
coordinates with respect to the vectors in g:
e ′ s = Je s = 〈Je s , w s 〉w s + 〈Je s , N〉N,
e ′ n = Je n = 〈Je n , w s 〉w s + 〈Je n , N〉N
and e ′ j = e j when j ∈ {1, 1, . . . , s − 1, s − 1, s, s + 1, s + 1, . . . , n − 1, n − 1, n}.
Then { ω
′
j = ω j , if j ≠ s, n,
and
⎧
⎨
⎩
ω ′ n
= 〈Je n , w s 〉ω˜s + 〈Je n , N〉ωñ
ω is ′ = 〈Je s , w s 〉ω i˜s + 〈Je s , N〉ω iñ , if i ≠ s, n
ω in ′ = 〈Je n , w s 〉ω i˜s + 〈Je n , N〉ω iñ , if i ≠ s, n
ω ij ′ = ω ij , if i, j ≠ s, n.
(3.3)
(3.4)
From now on, in order to simplify the notation, we omit the absolute value in densities.
The expression of dx (the density of S ∩ L s ), dL s and dL s[p] in terms of ω ′ is
dx = ω ′ 1 ∧ ω ′ 1 ∧ · · · ∧ ω′ s,
dL s = ω ′ s+1 ∧ ω ′ s+1 ∧ · · · ∧ ω′ n ∧ ω ′ n ∧
dL s[p] =
∧
ω ij
′
i=1,2,...,s
j=s+1,s+1,...,n,n
∧
ω ′ ij,
i=1,2,...,s
j=s+1,s+1,...,n,n
and the expression of dp (the density of S) in terms of ω is dp = ω 1 ∧ ω 1 ∧ · · · ∧ ω n .
On the other hand, by Lemma 3.1.1 it is satisfied
|〈Je n , w s 〉| = |〈Je s , N〉|. (3.5)
Indeed, vectors {e s , w s , e n , N} are an orthonormal basis of a 2-dimensional complex plane, the
orthogonal complement of the space generated by {e 1 , Je 1 , . . . , e s−1 , Je s−1 , e s+1 , Je s+1 , . . . , e n−1 , Je n−1 }.
By relations (3.3) and (3.5) we get
since ωñ vanishes on T S.
Then, by Lemma 3.1.3,
∫
M (s)
S∩L s≠∅
dx ∧ dL s = |〈Je n , w s 〉|dL s[p] ∧ dp = |〈JN, e s 〉|dL s[p] ∧ dp (3.6)
( ) 2s − 1 −1 ∫
r (S ∩ L s )dL s =
r
( ) 2s − 1 −1 ∫
=
r
S
S
∫
|〈JN, e s 〉|˜σ r (p)dL s[p] dp
L s[p]
∫
|〈JN, e s 〉|
L s[p]
|〈N, Je s 〉| r σ r(p)dL s[p] dp.
Note that in the last integrand we consider the absolut value in the denominator to be
sure that we consider the acute angle between the two intersecting subspace. This is not a

3.2 Integral of the r-th mean curvature integral 51
priori true since e s is any vector and, thus, not always the vector Je s (a normal vector to
S ∩ L s ⊂ L s ) has the desired orientation.
Now, we shall express dL s[p] in terms of dV ∧ de s where dV denotes the volume element of
the Grassmannian G C n−2,s−1 and de s the volume element of S 2n−2 . Fixed p, define the manifold
M = {(e s , V ) | e s ∈ T p S unit , V ∈ L s containing p and orthogonal to {N, JN, e s , Je s }},
which locally coincides with S 2n−2 × G C n,s−1 . Consider the fiber bundle
φ : M −→ L C s[p]
(e s , V ) ↦→ exp p {e s , Je s , V }
with fiber G C n,s. This is a double covering of L C s[p]
since vectors v and −v give the same complex
s-plane.
The pull-back of dL s[p] by the last mapping give the desired relation among the densities.
The expression of de s in terms of ω and the expression of dV in terms of ω ′ are
∧
de s =
ω sj ,
By (3.4) and (3.7) we have
∧
dL s[p] =
dV =
ω ij
′
i=1,2,...,s
j=s+1,s+1,...,n,n
= dV ∧
∧
i=1,2,...,s−1
j=1,1,...,s−1,s−1,˜s,s+1,s+1,...,n
∧
i=1,2,...,s−1
j=s+1,s+1,...,n−1,n−1
ω in ∧
∧
i=1,2,...,s
ω ′ ij. (3.7)
ω ′ in ∧
Next, we relate ∧ ω in ′ ∧ ω
′
in with ∧ ω sj . From (3.4) follows
∧
∧
ω in ′ = |〈Je n , w s 〉| s
i=1,2,...,s
i=1,2,...,s
∧
j=s+1,s+1,...,n−1,n
and also using that ω in ′ = ω′ in since ω′ in = 〈de′ i , e′ n〉 = 〈dJe ′ i , Je′ n〉 = ω ′ we obtain
i,n
∧
∧
ω in ′ = ω ′ in =
∧
〈Je n , w s 〉ω i˜s
In order to study
i=1,2,...,s−1
i=1,2,...,s−1
= |〈Je n , w s 〉| s−1 ∧
∧
i=1,1,...,s−1,s−1
i=1,2,...,s−1
i=1,2,...,s−1
ω i˜s ,
we use e s = Je s = 〈Je s , w s 〉w s + 〈Je s , N〉N and we obtain
∧
∧
ω is =
ω is
=
i=1,1,...,s−1,s−1
i=1,1,...,s−1,s−1
= 〈Je s , w s 〉 2(s−1) ∧
ω i˜s
.
ω i˜s
∧
i=1,1,...,s−1,s−1
i=1,1,...,s−1,s−1
ω i˜s .
ω is
ω sj .

52 Average of the mean curvature integral
Thus,
and
∧
i=1,1,...,s−1,s−1
ω i˜s = 〈Je s , w s 〉 −2(s−1)
∧
i=1,1,...,s−1,s−1
dL s[p] = |〈Je n, w s 〉| 2s−1
〈Je s , w s 〉 2(s−1) dV ∧ de s. (3.8)
Using |〈Je n , w s 〉| = |〈JN, e s 〉| and 〈Je s , w s 〉 2 = 1 − 〈JN, e s 〉 2 we get the result.
3.3 Mean curvature integral
First we give an expression for the integral over the space of complex r-planes of the integral
of the normal curvature in the direction JN (see (2.9)). By Example 2.4.20 we have that this
integral is a smooth valuation in CK n (ɛ). Moreover, it is not a multiple of the mean curvature
integral.
Theorem 3.3.1. Let S ⊂ CK n (ɛ) be compact oriented (possibly with boundary) hypersurface
of class C 2 oriented by a normal vector N, and s ∈ {1, . . . , n − 1}. Then
∫
)
˜kn
(∫S∩L (JÑ)dx dL s
s
L C s
= vol(G C n−2,s−1) ω ( )
2n−2 n −1 ( ∫
)
2sn − s − n
k n (JN) + (2n − 1)M 1 (S) ,
2s s n − s S
where ˜k n (JÑ) the normal curvature of S ∩ L s in the direction JÑ, k n(JN) the normal curvature
of JN in CK n (ɛ) and ω 2n−2 denotes the volume of the unit ball in the standard Euclidean
space of dimension 2n − 2.
Proof. Denote JÑ by e s.
By Lemma 3.1.3 we have ˜k n (JÑ) = k n(e s )
. Using equalities (3.6) and (3.8) we obtain
〈JN, e s 〉
I =
∫
=
∫L s
∫
S
S∩L s
∫ ∫RP 2n−2
˜ kn (JÑ)dxdL s
G C n−2,s−1
ω is
〈JN, e s 〉 2s
(1 − 〈JN, e s 〉 2 ) s−1 k n (e s )
〈JN, e s 〉 dV de sdp
As the integral over G C n−2,s−1 is independent of V , it follows
∫
I = vol(G C 〈JN, e s 〉
n−2,s−1)
S
∫RP 2s−1
2n−2 (1 − 〈JN, e s 〉 2 ) s−1 k n(e s )de s dp. (3.9)
In order to compute the integral over RP 2n−2 , we use polar coordinates and express the
normal curvature of e s in terms of the principal curvatures of T p S.
That is, if {f 1 , . . . , f 2n−1 } is an orthonormal basis of principal directions of T p S then
e s = ∑ 2n−1
j=1 〈e s, f j 〉f j , and
2n−1
∑
2n−1
∑
k n (e s ) = 〈e s , f j 〉 2 k n (f j ) = 〈e s , f j 〉 2 k j .
j=1
On the other hand, we consider a polar coordinates system θ 1 , θ 2 with respect to JN
defined by
|〈JN, e s 〉| = cos θ 1 , (3.10)
j=1

3.3 Mean curvature integral 53
and using spherical trigonometry,
〈e s , f j 〉 = cos θ 1 cos(JN, f j ) + sin θ 1 sin(JN, f j ) cos(e s , JN, f j )
= cos θ 1 cos α j + sin θ 1 sin α j cos θ 2 (3.11)
where cos(e s , JN, f j ) denotes the cosine of the spherical angle with vertex JN. Note that α j
are constants when the point is fixed.
Then, from the relations
Γ(s)Γ(n − s)
Γ(n + 1)
=
)
1 n −1
and
s(n − s)(
s
Γ(s)Γ(n − s + 1)
Γ(n + 1)
= 1 s
( n
s
) −1
,
we get
∫RP 2n−2 |〈JN, e s 〉| 2s−1
=
+
=
2n−1
∑
j=1
∫ π ∫S 2n−4
2n−1
∑
j=1
= ω 2n−2
2s
(1 − 〈JN, e s 〉 2 ) s−1 k n(e s )de s
k j
( ∫ S 2n−3 ∫ π/2
0
0
cos 2s−1 θ 1
sin 2s−2 θ 1
cos 2 θ 1 cos 2 α j sin 2n−3 θ 1 dθ 1 dS 2n−3 +
cos 2 θ 2 sin 2n−4 θ 2
∫ π/2
(
k j O 2n−3 cos 2 (2sn − s − n)Γ(s)Γ(n − s)
α j +
4(n − 1)Γ(n + 1)
( n −1 2n−1 ∑
( )
2sn − n − s
k j cos
s) 2 α j + 1 .
n − s
j=1
Integrating over S and using
we obtain the stated result.
0
k n (JN) =
2n−1
∑
j=1
cos 2s−1 θ
)
1
sin 2s−2 sin 2 θ 1 sin 2 α j sin 2n−3 θ 1 dθ 1 dθ 2 dS 2n−4 + 0
θ 1
)
k j 〈JN, f j 〉 2 =
Γ(s)Γ(n − s + 1)
4(n − 1)Γ(n + 1)
2n−1
∑
j=1
k j cos 2 α j
Theorem 3.3.2. Let S ⊂ CK n (ɛ) be a compact (possibly with boundary) hypersurface of class
C 2 oriented by a normal vector N, and let s ∈ {1, ..., n − 1}. Then
∫
L C s
M (s)
1 (S∩L s)dL s = ω 2n−2vol(G C n−2,s−1 ) ( n −1 (
(2n − 1)
2s(2s − 1) s) 2ns − n − s ∫
M 1 (S) +
n − s
where k n (JN) denotes the normal curvature in the direction JN ∈ T S.
L C s
S
)
k n (JN)
Proof. By Proposition 3.2.2 and Lemma 3.1.4 we have
∫
M (s)
1 (S ∩ L s)dL s = 1 ∫ ∫
|〈JN, e s 〉|
2s − 1
∫RP 2s−1
2n−2 (1 − 〈JN, e s 〉 2 ) s−1 σ 1(p; e s , V )dV de s dp
= 1
2s − 1
∫
S
= vol(GC n−2,s−1 )
2s − 1
S
S
G C n−2,s−1
|〈JN, e s 〉|
∫RP 2s−1 ∫
2n−2 (1 − 〈JN, e s 〉 2 ) s−1 (tr(II| V ) + II(e s , e s ))dV de s dp
G C n−2,s−1
∫
|〈JN, e s 〉|
∫RP 2s−1 ( )
s − 1
2n−2 (1 − 〈JN, e s 〉 2 ) s−1 n − 2 tr(II| E) + k n (e s ) de s dp,
where E = 〈N, JN, e s , Je s 〉 ⊥ .

54 Average of the mean curvature integral
Note that if s = 1 then dim V = 0. Although the integral ∫ G
tr(II|V )dV has no
C
n−2,s−1
sense, last equality above remains true since s−1
n−2 trII| E = 0.
We shall study the following integrals
J E = vol(GC n−2,s−1 )
2s − 1
J s = vol(GC n−2,s−1 )
2s − 1
∫
s − 1
n − 2
∫
S
S
∫RP 2n−2 |〈JN, e s 〉| 2s−1
(1 − 〈JN, e s 〉 2 ) s−1 tr(II| E)de s dp
∫RP 2n−2 |〈JN, e s 〉| 2s−1
(1 − 〈JN, e s 〉 2 ) s−1 k n(e s )de s dp.
The second integral is the same (except for a constant factor) as the integral (3.9) in Proposition
3.3.1. Thus, we know
J s = O 2n−3vol(G C n−2,s−1 ) ( ) n −1 ( ∫
)
2sn − s − n
k n (JN) + (2n − 1)M 1 (S) .
4s(2s − 1)(n − 1) s n − s S
In order to study the integral J E , we shall use polar coordinates in the same way and with
the same notation as in the proof of Theorem 3.3.1. Let {e 1 , Je 1 , . . . , e s−1 , Je s−1 } be a J-basis
of E ∩ T p L s and let {e s+1 , Je s+1 , . . . , e n−1 , Je n−1 } be a J-basis of E ∩ (T p L s ) ⊥ . With respect
to this orthonormal basis of E
tr(II| E ) =
n−1
∑
i=1,i≠s
(k n (e i ) + k n (Je i )).
If we denote by {f 1 , . . . , f 2n−1 } a basis of principal directions of S at p, we obtain
k n (e i ) =
j=1
2n−1
∑
j=1
k n (f j )〈e i , f j 〉 2 =
2n−1
∑
and using polar coordinates with respect to JN, we get
⎛
⎞
2n−1
∑
n−1
∑
tr(II| E ) = k j
⎝ (〈e i , f j 〉 2 + 〈Je i , f j 〉 2 ) ⎠
=
2n−1
∑
j=1
k j
⎛
⎝
i=1,i≠s
n−1
∑
i=1,i≠s
j=1
k j 〈e i , f j 〉 2 ,
⎞
(cos 2 (e i , JN, f j ) + cos 2 (Je i , JN, f j ) ⎠ .
We denote by (u, v, w) the spherical angle with vertex v and sides on u and w, cos φ ij =
cos(e i , JN, f j ) and cos φ ij
= cos(Je i , JN, f j ). Then, to study the integral J E we have to deal
with the following integral
∫S 2n−3 ∫ π/2
0
cos 2s−1 θ 1
sin 2s−2 θ 1
sin 2 α j sin 2n−3 θ 1·
· (cos 2 φ 1j + · · · + cos 2 φ s−1,j + cos 2 φ s+1,j + · · · + cos 2 φ n−1,j )dθ 1 dS 2n−3
= sin 2 Γ(s)Γ(n − s)
α j ·
2Γ(n)
∫
· (cos 2 φ 1j + · · · + cos 2 φ s−1,j + cos 2 φ s+1,j + · · · + cos 2 φ n−1,j )dS 2n−3
S 2n−3
where θ 1 and α i are defined in (3.10) and (3.11).

3.3 Mean curvature integral 55
Denote by ˜S 2n−1 the subset of S 2n−1 defined by all points not in span{N, JN}. Consider
the well-defined map
Π : ˜S2n−1 −→ {N, JN} ⊥
v ↦→ proj {N,JN} ⊥ (v)
||proj {N,JN} ⊥(v)||
Note that Π(e a ) = e a , with a ∈ {1, 1, . . . , s − 1, s − 1, s + 1, s + 1, . . . , n − 1, n − 1}
Then, V ⊥ inside {N, JN} ⊥ is
E ⊥ ∩ {N, JN} ⊥ = ({N, JN, e s , Je s } ⊥ ) ⊥ ∩ {N, JN} ⊥ = {Π(e s ), JΠ(e s )}. (3.12)
As cos(φ aj ) = cos(e a , JN, f j ) denotes the cosine of the spherical angle with vertex JN and
points in e a and f j , by definition, it coincides with 〈Π(e a ), Π(f j )〉 = 〈e a , Π(f j )〉. Then, as Π(f j )
is a unit vector contained in the vector subspace with basis {e 1 , Je 1 , . . . , e s−1 , Je s−1 , Π(e s ), JΠ(e s )}
it is satisfied
1 = 〈Π(f j ), Π(f j )〉 2
= 〈e 1 , Π(f j )〉 2 + · · · + 〈Je s−1 , Π(f j )〉 2 + 〈Π(e s ), Π(f j )〉 2 + 〈JΠ(e s ), Π(f j )〉 2
.
and we get
∫
(cos 2 φ 1j + · · · + cos 2 φ s−1,j + cos 2 φ s+1,j + ... + cos 2 φ n−1,j )dS
S∫
2n−3
= (1 − 〈Π(e s ), Π(f j )〉 2 − 〈JΠ(e s ), Π(f j )〉 2 )dS.
S 2n−3
Now, we use polar coordinates θ 2 , θ 3 with respect to Π(f j ) such that
〈Π(e s ), Π(f j )〉 = cos θ 2 , θ 2 ∈ (0, π),
and
〈JΠ(e s ), Π(f j )〉 = sin(Π(e s ), Π(f j )) cos(Π(e s ), Π(f j ), JΠ(f j )) = sin θ 2 cos θ 3 , θ 3 ∈ (0, π).
By Lemma 3.1.2 and the relation
π
O 2n−3 = O 2n−5
n − 2
we have
∫ π ∫ π
∫S 2n−5
0
= O 2n−3 −
(1 − cos 2 θ 2 − sin 2 θ 2 cos 2 θ 3 ) sin 2n−4 θ 2 sin 2n−5 θ 3 dθ 3 dθ 2 dS 1
0
∫ π ∫S 2n−4
0
∫ π
cos 2 θ 2 sin 2n−4 θ 2 −
∫S 2n−5
= O 2n−3 − O √ √
2n−3
πΓ(n − 2) πΓ(n −
1
2n − 2 − O 2n−52
4Γ(n − 1 2 ) Γ(n)
( π
= O 2n−5
n − 2 − π
2(n − 1)(n − 2) − π
2(n − 1)(n − 2)
O 2n−5 π
=
(2n − 4)
2(n − 1)(n − 2)
= O 2n−5π
2(n − 1) .
0
∫ π
cos 2 θ 3 sin 2n−5 θ 3 sin 2n−2 θ 2
2 )
)
0

56 Average of the mean curvature integral
Thus,
J E = O 2n−3vol(G C (
n−2,s−1 )n(s − 1) n −1 (
∫ )
(2n − 1)M 1 (S) − k n (JN)
2s(2s − 1)(n − s)(n − 1) s)
S
and adding both expressions of J E and J s we get the result
J E + J s = O 2n−3vol(G C n−2,s−1 ) ( ) n −1
·
4s(2s − 1)(n − 1) s
(
n(s − 1)
· ((2n − 1) + (2n − 1)
n − s )M 1(S) + ( 2sn − n − s
)
n(s − 1)
−
n − s n − s
∫S
) k n (JN)
= O 2n−3vol(G C n−2,s−1 ) ( ) n −1
·
4s(2s − 1)(n − 1) s
( ∫ )
2n − 1
·
n − s (n − s + ns − n)M 2sn − n − s − ns + n
1(S) + k n (JN)
n − s
S
= O 2n−3vol(G C n−2,s−1 ) ( ) n −1 ( ∫ )
s(2n − 1)(n − 1)
M 1 (S) + k n (JN) .
4s(2s − 1)(n − 1) s
n − s
S
It is natural to ask which functionals we have to integrate over the space of complex s-planes
to obtain the mean curvature integral of the initial hypersurface.
Theorem 3.3.3. Let S ⊂ CK n (ɛ) be a compact (possibly with boundary) oriented hypersurface
of class C 2 . If we define
ν(S) = (2ns − n − s)M 1 (S) − n − s ∫
k n (JN)dx,
2s − 1
S
then
∫
L C r
ν(S ∩ L r )dL r = ω 2n−2vol(G C n−2,s−1 )2(n − 1)(2n − 1)(s − 1)
M 1 (S).
(2s − 1)
Proof. The result follows straightforward from Theorems 3.3.1 and 3.3.2.
3.4 Reproductive valuations
Definition 3.4.1. Suppose given, for each s ∈ N, a valuation in CK n (ɛ), ϕ (s) . It is said that
the collection {ϕ (s) } satisfies the reproductive property if for any regular domain Ω,
∫
ϕ (s) (Ω ∩ L s )dL s = c n,s ϕ(Ω),
for some constant c n,s depending on n and s.
L C s
Remark 3.4.2. Recall that mean curvature integral for regular domains extend to all K(C n ).
Also ∫ ∂Ω k n(JN) extends to K(C n ) since it coincides with Γ 2n−2,n−1 (Ω) (cf. Example 2.4.20).
As neither the mean curvature integral M (s)
1 (∂Ω ∩ L s), nor the integral of the normal
curvature, ∫ ∂Ω∩L s
k n (JÑ)dx, satisfy the reproductive property, it is natural to ask whether
there exists some linear combination of these such that satisfies this property. We consider a
linear combination since, in C n , they constitute a basis of Val U(n)
n−2 (Cn ).

3.4 Reproductive valuations 57
Theorem 3.4.3. Let Ω ⊂ CK n (ɛ) be a regular domain, s ∈ {1, ..., n−1}. Consider the smooth
valuations defined by
∫
ϕ 1 (Ω) = M 1 (∂Ω) − k n (JN)
∂Ω
and
Then,
and
∫
L C s
∫
ϕ 2 (Ω) = (2s − 1)(2n − 1)M 1 (∂Ω) + k n (JN).
∂Ω
ϕ 1 (Ω ∩ L s )dL s = ω 2n−2vol(G C ( )
n−2,s−1 )(s − 1)(2n − 1) n −1
ϕ 1 (Ω)
(2s − 1)(n − s)
s
∫
L C s
ϕ 2 (Ω ∩ L s )dL s = ω 2n−2vol(G C n−2,s−1 )
(2s − 1)
( ) n − 2 −1
ϕ 2 (Ω).
s − 1
In C n , each of ϕ 1 (Ω) and ϕ 2 (Ω) expands a 1-dimensional subspace of reproductive valuations
of degree 2n − 2.
Proof. Let
∫
ν(Ω) = aM 1 (∂Ω) + b k n (JN).
∂Ω
We look for relations between a and b to be ν a reproductive valuation, that is,
∫
ν(Ω ∩ L s )dL s = λν(Ω).
L C s
From Theorems 3.3.1 and 3.3.2 we have
∫
∫ (
)
ν(Ω ∩ L s )dL s = aM 1 (∂Ω ∩ L s ) + b k n
∫∂Ω∩L (JÑ) s
L C s
L C s
= ω 2n−2vol(G C n−2,s−1 ) ( ) n −1((
b(2s − 1)(2sn − n − s)
) ∫
a + k n (JN)+
2s(2s − 1) s
n − s
∂Ω
+ ( a(2ns − n − s)
+ b(2s − 1) ) )
(2n − 1)M 1 (∂Ω) .
n − s
Thus, ν is reproductive if and only if for some λ ∈ R
( )
a(2ns − n − s)
(2n − 1)
+ b(2s − 1) = λa,
n − s
b(2s − 1)(2sn − n − s)
a + = λb.
n − s
Solving this system we get two solutions, a = −b, λ =
1)(2n − 1), λ =
2n(n − 1)
.
n − s
2s(s − 1)(2n − 1)
n − s
and a = b(2s −
Remark 3.4.4. Last theorem gives all valuations in Val U(n)
n−2 (Cn ) such that they satisfy the
reproductive property. Why are these valuations reproductive Are they special in some
sense It shall be interesting to know the answer and also to have a geometric interpretation
for these valuations.

58 Average of the mean curvature integral
3.5 Relation with some valuations defined by Alesker
In Section 2.3 we recalled the definition of valuations U k,p in C n . They constitute a basis for
Val U(n) (C n ). From this basis it can be established the following theorem by Alesker
Theorem 3.5.1. (Theorem 3.1.2 [Ale03]) Let Ω be a regular domain in C n . Let 0 < q <
n, 0 < 2p < k < 2q. Then,
∫
L C q
U k,p (Ω ∩ L q )dL q =
where constants γ p depend only on n, q and p.
[k/2]+n−q
∑
p=0
γ p · U k+2(n−q),p (Ω),
In the following theorem we give the constants γ p with arbitrary n, q and k = 2q − 2.
Theorem 3.5.2. Let Ω be a regular domain and 0 < q < n. Then,
∫
U 2q−2,p (Ω ∩ L q )dL q = ω 2q−2ω 2n−2 vol(G C n−2,q−1 )vol(GC q−2,q−p−1 )
L C (q − p)(2q − 2p − 1) ( )(
n−2 q−2
) ·
q
q−1 q−p−1
( )
(2n − 3)(n − 1)(n − q + p)
·
U 2n−2,1 (Ω) − (2n − 1)n(n − q + p − 1)U 2n−2,0 (Ω) .
ω 2n−2
First, we express ∫ ∂Ω k n(JN) (a translation invariant continuous valuation) in terms of
{U k,p }.
Proposition 3.5.3. Let Ω be a regular domain in C n . Then
∫
k n (JN)dp = n(2n − 3)(2n − 2)2 ω 2
U 2n−2,1 (Ω) − 2n(2n − 1)(2n 2 − 4n + 1)ω 2 U 2n−2,0 (Ω).
ω 2n−2
∂Ω
Proof. From the relations among valuations {U k,p } and mean curvature integrals (see (2.3))
we have
U 2n−2,0 (Ω) = 1 M 1 (∂Ω) (3.13)
2nω 2
and
∫
1
U 2n−2,1 (Ω) =
M 1 (∂Ω ∩ L n−1 )dL n−1 .
(2n − 2)ω 2
L C n−1
Using Proposition 3.3.2 with s = n − 1, we get the result.
Proof of the theorem 3.5.2. From the definition of U k,p we have
∫
1
U k,p (Ω) =
M k−2p (Ω ∩ L n−p )dL n−p
2(n − p)ω 2n−k
and from Theorem 3.3.2
∫
1
U 2q−2,p (Ω ∩ L q ) =
2(q − p)ω 2
L C q−p
G C n,n−p
M 1 ((∂Ω ∩ L q ) ∩ L q−p )dL q−p
) −1·
O 2q−3 vol(G C q−2,q−p−1
=
) ( q
8(q − p) 2 (q − 1)(2q − 2p − 1)ω 2 q − p
(
)
2q(q − p) − 2q + p
· (2q − 1) M 1 (∂Ω ∩ L q ) + k n
p
∫∂Ω∩L (JÑ) q
where Ñ denotes the normal inward vector field to ∂Ω ∩ L q as a hypersurface in L q . Using
again Theorems 3.3.1 and 3.3.2, we express the integrals over L C q as an integral oer ∂Ω. Finally,
from the relation in Proposition 3.5.3 and (3.13) we get the result.

3.6 Example: sphere in CK 3 (ɛ) 59
3.6 Example: sphere in CK 3 (ɛ)
In this section we check Theorem 3.3.2 in the case of a sphere of radius R in CK 3 (ɛ).
On page 18 we give an expression for the principal curvatures of a sphere of radius R in
CK n (ɛ). Using this expression we get
∫
and
k n (JN) = 2 cot ɛ (2R)V (∂B R ) = 2 cos2 ɛ(R) + sin 2 ɛ(R)
∂B R
2 sin ɛ (R) cos ɛ (R)
π 3
3! 6 sin5 ɛ(R) cos ɛ (R)
= π 3 (cos 2 ɛ(R) + sin 2 ɛ(R)) sin 4 ɛ(R) = π 3 (2 sin 6 ɛ(R) + sin 4 ɛ(R))
= π 3 (2 cos 6 ɛ(R) − 5 cos 4 ɛ(R) + 4 cos 2 ɛ(R) − 1)
M 1 (∂B R ) = π3 6
3!4 (5 sin4 ɛ(R) cos 2 ɛ(R) + sin 6 ɛ(R)) = π3
4 (6 sin6 ɛ(R) + 5 sin 4 ɛ(R))
= π 3 ( 6 5 cos6 ɛ(R) − 13
5 cos4 ɛ(R) + 8 5 cos2 ɛ(R) − 1 5 ).
Thus, the right hand side of Theorem 3.3.2 is
O 3 π 3
144 ((42 + 2) cos6 ɛ(R) − (13 · 7 + 5) cos 4 ɛ(R) + (56 + 4) cos 2 ɛ(R) − (7 + 1))
( 11
= 2π 5 36 cos6 ɛ(R) − 2 3 cos4 ɛ(R) + 5
12 cos2 ɛ(R) − 1 )
.
18
The left hand side of Theorem 3.3.2 is
∫
M (2)
1 (∂B R ∩ L 2 )dL 2 .
L C 2
Let us compute first M (2)
1 (∂B R ∩ L 2 ), for a fixed complex 2-plane, L 2 . Recall that the intersection
between a sphere and L 2 is a sphere inside L 2 with radius r satisfying cos ɛ (R) =
cos ɛ (r) cos ɛ (ρ) where ρ is the distance from the origin of the sphere B R to the plane L 2 (cf.
[Gol99, Lemma 3.2.13]).
Thus,
∫
L C 2
= 1 12
M (2)
1 (∂B R ∩ L 2 ) = 1 ∫
(2 cot ɛ (r) + 2 cot ɛ (2r))
3 ∂B R
M (2)
1 (∂B R ∩ L 2 )dL 2
∫
G C 3,1
∫S 1 ∫ R
= 2πV (GC 3,1 )2
12
= πV (GC 3,1 )
3
0
∫ R
0
= 2 3 (cot ɛ(r) + cot ɛ (2r)) 4π2 sin 3
2!
ɛ(r) cos ɛ (r)
= 1
12 (4 cos4 ɛ(r) − 5 cos 2 ɛ(r) + 1)
= 1 1
12 cos 4 ɛ(R) (4 cos4 ɛ(R) − 5 cos 2 ɛ(R) cos 2 ɛ(ρ) + cos 4 ɛ(ρ)).
cos 4 ɛ(ρ)
cos 4 ɛ(ρ) (4 cos4 ɛ(R) − 5 cos 2 ɛ(R) cos 2 ɛ(ρ) + cos 4 ɛ(ρ))2 cos ɛ (ρ) sin ɛ (ρ)
(4 cos 4 ɛ(R) − 5 cos 2 ɛ(R) cos 2 ɛ(ρ) + cos 4 ɛ(ρ))2 cos ɛ (ρ) sin ɛ (ρ)
( 11
12 cos6 ɛ(R) − 2 cos 4 ɛ(R) + 5 4 cos2 ɛ(R) − 1 6
and we get the same result in both side of the expression in Theorem 3.3.2.
)

Chapter 4
Gauss-Bonnet Theorem and Crofton
formulas for complex planes
In this chapter we obtain an expression for the measure of complex r-planes intersecting a
compact domain in CK n (ɛ). That is, we give an expression of
∫
χ(Ω ∩ L r )dL r (4.1)
L C r
for a regular domain Ω ⊂ CK n (ɛ) as a linear combination of the so-called Hermitic intrinsic
volumes valuations in CK n (ɛ) (cf. Definition 2.4.11). The method we use consists on computing
the variation, when the domain moves along the flow induced by a smooth vector field, of the
measure of complex r-planes intersecting the convex domain. From the theory of valuations on
C n , we know that the expression is a linear combination of certain valuations. Thus, computing
also the variation of these valuations and then comparing both results we shall deduce the final
expression.
Using the same method we obtain an expression (cf. Theorem 4.4.1) for the Euler characteristic
of a compact domain in terms of its Gauss curvature of the boundary, its volume
and others Hermitian intrinsic volumes. This expression is analogous to the one obtained in
[San04, page 309] for real space forms.
Relating these two results we shall obtain another expression for the Euler characteristic.
This one involves the measure of complex hyperplanes intersecting the regular domain (cf.
Theorem 4.4.5).
4.1 Variation of the Hermitian intrinsic volumes
The study of the variation of a valuation when the domain moves along the flow of a smooth
vector field will be useful to deduce some properties of the valuation. In [BF08] it is given
the variation of some valuations on C n and this variation is used to characterize monotone
valuations. In this work, we give the variation of the Hermitian intrinsic volumes (cf. Definition
2.4.11) on CK n (ɛ) and we use it to deduce expression (4.1) in terms of these valuations.
In order to obtain the variation of Hermitian intrinsic volumes on CK n (ɛ) we follow the
same method as in the proof of Corollary 2.6 in [BF08]. First, we recall the definition of the
Rumin derivative, introduced in [Rum94].
Definition 4.1.1. Let µ ∈ Ω 2n−1 (S(CK n (ɛ))), let α be the contact form of S(CK n (ɛ)) and let
α ∧ ξ ∈ Ω 2n−1 (S(CK n (ɛ))) be the unique (cf. [Rum94]) form such that d(µ + α ∧ ξ) is multiple
of α. Then, the Rumin operator D is defined as
Dµ := d(µ + α ∧ ξ).
61

62 Gauss-Bonnet Theorem and Crofton formulas for complex planes
Let us recall the definition of the Reeb vector field over a contact manifold.
Definition 4.1.2. Let M be a contact manifold with contact form α. The Reeb vector field
T is the only vector field over M such that
{
iT α = 1,
L T α = 0.
(4.2)
If the contact manifold is the fiber tangent bundle of a Riemann manifold, then the Reeb
vector field coincides with the geodesic flow (cf. [Bla76, page 17]). This is the situation in this
work, we consider the unit tangent bundle of CK n (ɛ) (cf. Lemma 1.3.7 and Remark 1.3.8).
Note that the condition L T α = 0 is equivalent to i T dα = 0. Indeed,
L T α = i T dα + d(i T α) = i T dα = dα(T ).
The following lemma contains the value of the contraction of T with α, β, γ and θ i defined
in Section 2.4.2.
Lemma 4.1.3. In S(CK n (ɛ)) it is satisfied
i T α = 1, i T θ 1 = γ,
i T θ 2 = β, i T β = i T γ = i T θ 0 = i T θ s = 0.
Proof. The first equality is a characterization of the Reeb vector field. Moreover, i T θ s =
−i t dα = di T α − L T α = 0.
As T is the geodesic flow, it satisfies α i (T ) = β i (T ) = 0 and α 1i = β 1i = 0, i ∈ {2, ..., n}.
We get the result using Definition (2.6) extended to CK n (ɛ).
In [BF08] it is proved the following lemma, which allow to calculate the variation of a
valuation defined from an invariant smooth form. The result is proved in C n but the same
remains true for ɛ ≠ 0, and for any Riemann manifold. Here we repeat the proof in detail for
CK n (ɛ).
Lemma 4.1.4 ([BF08] Lemma 2.5). Suppose that Ω ⊂ CK n (ɛ) is a regular domain, N the
outward unit vector field to ∂Ω, X is a smooth vector field on CK n (ɛ) with flow F t and µ a
smooth valuation given by a (2n − 1)-form ρ in S(CK n (ɛ)). Then
∫
d
dt∣ µ(F t (Ω)) = δ X µ(Ω) =
t=0
N(Ω)
〈X, N〉 i T (Dβ)
where T is the Reeb vector field of S(CK n (ɛ)) and Dρ is the Rumin operator of ρ.
Proof. Let ˜X be a lift of X at S(CK n (ɛ)) such that it preserves α, i.e. L ˜X(α) = 0. Then, if
˜F denotes the flow of ˜X

4.1 Variation of the Hermitian intrinsic volumes 63
δ X µ(Ω) = d ( ∫ )
dt∣ β = d ( ∫ )
∗ t=0 N(F t(Ω)) dt∣ ˜F t β
t=0 N(Ω)
∫
=
(1)
=
(3)
=
(4)
=
(5)
=
(6)
=
(7)
=
L ˜Xβ
N(Ω)
∫
∫
∫
∫
∫
∫
∫
=
N(Ω)
N(Ω)
N(Ω)
N(Ω)
N(Ω)
N(Ω)
N(Ω)
∫
(2)
i ˜Xdβ + d(i ˜Xβ) = i ˜Xdβ
N(Ω)
i ˜XDβ − i ˜Xd(α ∧ η)
i ˜XDβ
i ˜X(α ∧ ρ)
(i ˜Xα)ρ
α( ˜X)i T (α ∧ ρ)
〈X, N〉i T Dβ
First, note that we can change the variation of F t (Ω) in the first integral for the Lie derivative
of the integrated form since N(F t (Ω)) = ˜F t (N(Ω)).
For (1) and (2), we use the following property of the Lie derivative, L ˜Xβ
= i ˜Xdβ + d(i ˜Xβ),
and that the second term is an exact form, thus the integral vanishes.
Equality (3) follows directly from the definition of the Rumin operator.
For (4) we use
i ˜Xd(α ∧ η) = L ˜X(α ∧ η) − d(i ˜X(α ∧ η))
and that the second term is an exact form. The first term can be rewritten as
L ˜X(α ∧ η) = (L ˜Xα) ∧ η + α ∧ L ˜Xη,
and so, its integral vanishes since ˜X preserves α, which vanishes over the normal fiber bundle
(cf. Lemma 1.3.12).
As the Rumin operator is, by definition, a 2n-form multiple of α, and it is defined on the
normal fiber bundle, we get (5).
For (6), using the notion of contraction we get
i ˜X(α ∧ ρ) = (i ˜Xα) ∧ ρ + α ∧ (i ˜Xρ).
The second term vanishes over N(Ω).
To get equality (7), we repeat the same argument as in (4) in order to obtain the form
α ∧ ρ, which is the Rumin operator of β.
Finally, we recall the definition of α and that the integral is over the unit fiber normal
bundle, so that the points are (x, N) with x ∈ ∂Ω and N the unit normal vector on ∂Ω at
x.
The previous lemma allows us to compute the variation of any valuation given by a form,
once we know its Rumin operator. In this chapter we give the variation of the Hermitian
intrinsic volumes in CK n (ɛ) (in [BF08] is given for ɛ = 0).
In the following lemma we give the derivative β k,q and γ k,q (defined in 2.4.6) using Lemma
2.4.8. They will be used for the computation of the Rumin operator.

64 Gauss-Bonnet Theorem and Crofton formulas for complex planes
Lemma 4.1.5. In CK n (ɛ)
and
dβ k,q = c n,k,q (θ n−k+q
0 ∧ θ k−2q
1 ∧ θ q 2 − ɛ(n − k + q)α ∧ β ∧ θn−k+q−1 0 ∧ θ k−2q
1 ∧ θ q 2 )
dγ k,q =c n,k,q (θ n−k+q
0 ∧ θ k−2q
1 ∧ θ q 2 − ɛθn−k+q−1 0 ∧ θ k−2q
1 ∧ θ q+1
2 − ɛα ∧ β ∧ θ n−k+q−1
0 ∧ θ k−2q
1 ∧ θ q 2
(n − k + q − 1)
− ɛ α ∧ γ ∧ θ n−k+q−2
0 ∧ θ k−2q+1
1 ∧ θ q 2
2
(n − k + q − 1)
− ɛ β ∧ γ ∧ θ 01 ∧ θ n−k+q−2
0 ∧ θ k−2q
1 ∧ θ q 2
2
).
From the previous lemma and following the method used by [BF08] we can compute the
variation of B k,q and Γ k,q in CK n (ɛ).
Notation 4.1.6. We denote
∫
˜B k,q = ˜B k,q (Ω) :=
∂Ω
∫
〈X, N〉β k,q and ˜Γk,q = ˜Γ k,q (Ω) :=
∂Ω
〈X, N〉γ k,q .
Proposition 4.1.7. Let X be a smooth vector field defined on CK n (ɛ) and Ω ⊂ CK n (ɛ) be
a regular domain. The variation in CK n (ɛ) of valuations B k,q and Γ k,q with respect to X is
given by
and
δ X B k,q (Ω) = 2c n,k,q (c −1
n,k−1,q (k − 2q)2˜Γk−1,q − c −1
n,k−1,q−1 (n + q − k)q˜Γ k−1,q−1
+c −1
n,k−1,q−1 (n + q − k + 1 2 )q ˜B k−1,q−1 − c −1
n,k−1,q (k − 2q)(k − 2q − 1) ˜B k−1,q
+ɛ(c −1
n,k+1,q+1 (k − 2q)(k − 2q − 1) ˜B k+1,q+1 − c −1
n,k+1,q (n − k + q)(q + 1 2 ) ˜B k+1,q ))
δ X Γ k,q (Ω) = 2c n,k,q
(
c −1
n,k−1,q (k − 2q)2˜Γk−1,q − c −1
n,k−1,q−1 (n + q − k)q˜Γ k−1,q−1
+c −1
n,k−1,q−1 (n + q − k + 1 2 )q ˜B k−1,q−1 − c −1
n,k−1,q (k − 2q)(k − 2q − 1) ˜B k−1,q
+ɛ ( c −1
n,k+1,q+1 2(k − 2q)(k − 2q − 1) ˜B k+1,q+1 − c −1
n,k+1,q ((n − k + q)(2q + 3 2 ) − 1 2 (q + 1)) ˜B k+1,q
−c −1
n,k+1,q+1 (k − 2q)2˜Γk+1,q+1 + c −1
n,k+1,q (n − k + q − 1)(q + 1)˜Γ k+1,q
−ɛ(c −1
n,k+3,q+2 (k − 2q)(k − 2q − 1) ˜B k+3,q+2 − c −1
n,k+3,q+1 (n − k + q − 1)(q + 3 2 ) ˜B k+3,q+1 ) )) .
Proof. We first study the valuation given by β k,q .
Lemma 4.1.4 provides an expression for the variation of a smooth valuation. In order to
use this lemma, it is enough to find i T Dβ k,q and i T γ k,q modulo α and dα since the latter forms
vanish over N(Ω) (cf. Lemma 1.3.12).
We will use the following fact from the proof of Proposition 4.6 in [BF08]: for max{0, k −
n} ≤ q ≤ k/2 < n there exists an invariant form ξ k,q ∈ Ω 2n−1 (S(C n )) such that
and
dα ∧ ξ k,q ≡ −θ n−k+q
0 θ k−2q
1 θ q 2 mod(α), (4.3)
ξ k,q ≡ βγθ n+q−k−1
0 θ k−2q−2
1 θ q−1
2 (4.4)
∧ ( (n + q − k)qθ1 2 )
− (k − 2q)(k − 2q − 1)θ 0 θ 2 mod(α, dα).

4.1 Variation of the Hermitian intrinsic volumes 65
In order to find δ X B k,q for general ɛ, we take a form ξ ɛ ∈ Ω 2n−1 (S(CK n (ɛ))) such that ξ ɛ (p,v) ≡
ξ (p ′ ,v ′ ) when we identify T (p,v) S(CK n (ɛ)) and T (p ′ ,v ′ )C n , for every (p, v) ∈ S(CK n (ɛ)), (p ′ , v ′ ) ∈
S(C n ). That is, as ξ ɛ is an invariant form, it can be expressed as a linear combination of
products with the forms α, β, θ 0 , θ 1 , θ 2 and θ s . We take this expression as a definition of ξ ɛ .
From Lemma 4.1.5 and (4.3) we have that d(β k,q + c n,k,q α ∧ ξ ɛ ) ≡ 0 modulo α.
By Lemma 2.4.8, the exterior differential of ξ ɛ is
dξ ɛ ≡ θ n+q−k−1
0 θ k−2q−2
1 θ q−1
2 ((n − k + q)qθ1 2 − (k − 2q)(k − 2q − 1)θ 0 θ 2 )
∧ (γθ 1 − 2βθ 0 + 2ɛβθ 2 ) mod(α, dα)
and the contraction of dβ k,q with respect to the vector field T , by Lemma 4.1.3, is
i T dβ k,q ≡ c n,k,q θ n+q−k−1
0 θ k−2q−1
1 θ q−1
2
∧ ((k − 2q)γθ 0 θ 2 + qβθ 0 θ 1 − ɛ(n − k + q)βθ 1 θ 2 )
mod(α).
By substituting the last expressions in i T Dβ k,q ≡ i T dβ k,q − c n,k,q dξ ( mod α, dα), we get
the result.
The variation of Γ k,q with k ≠ 2q can be obtained using the relation among Γ k,q and B k,q
given in Proposition 2.4.8 and the variation of B k,q .
To compute δ X Γ 2q,q , note that dγ 2q,q has 3 terms not multiple of α (cf. Lemma 4.1.5). As
before we take ξ ɛ 1 , ξɛ 2 ∈ Ω2n−1 (S(CK n (ɛ))) corresponding to ξ 2q,q , and ξ 2q+2,q+1 respectively.
Let us consider also
ξ3 ɛ = n − q − 1 βγθ n−q−2
0 θ q 2
2
. (4.5)
Then the Rumin differential of γ 2q,q is given by Dγ 2q,q = d(γ 2q,q + c n,2q,q α ∧ (ξ1 ɛ − ɛξɛ 2 − ɛξɛ 3 )).
Indeed, dα ∧ ξ1 ɛ cancels the first term of dγ 2q,q modulo α, and dα ∧ ξ2 ɛ cancels the second one.
The third term is canceled exactly by dα ∧ ξ3 ɛ.
Now, using Lemmas 4.1.5 and 4.1.3 we get
i T dγ 2q,q ≡qβθ n−q
0 θ q−1
2 − ɛ(q + 2)βθ n−q−1
and from (4.4) and (4.5)
0 θ q 2 − ɛn − q − 1
2
γθ n−q−2
0 θ 1 θ q 2 mod(α, dα),
dξ ɛ 1 ≡ (n − q)qθ n−q−1
0 θ q−1
2 (γθ 1 − 2βθ 0 + 2ɛβθ 2 ) mod(α, dα).
dξ ɛ 2 ≡ (n − q − 1)(q + 1)θ n−q−2
0 θ q 2 (γθ 1 − 2βθ 0 + 2ɛβθ 2 ) mod(α, dα).
dξ3 ɛ ≡ n − q − 1 θ n−q−2
0 θ q 2
2
(γθ 1 − 2βθ 0 + 2ɛβθ 2 ) mod(α, dα).
Plugging this into i T Dγ 2q,q ≡ i T dγ 2q,q − c n,2q,q (dξ1 ɛ − ɛdξɛ 2 − ɛdξɛ 3 ) mod (α, dα) gives the result.
Remark 4.1.8. For ɛ = 0 the variation of B k,q coincides with the variation of Γ k,q and we get
the result of Proposition 4.6 in [BF08].
From the previous proposition we can obtain easily the variation of the Gauss curvature
integral. We know that this variation vanishes in C n , for the Gauss-Bonnet theorem, but not
in the other complex space forms.
Corollary 4.1.9. In CK n (ɛ) the variation of the Gauss curvature integral is
δ X M 2n−1 (∂Ω) = 2ɛω 2n−1 (2(n − 1)˜Γ 1,0 − (3n − 1) ˜B 1,0 + 3
2π ɛ(2n − 1) ˜B 3,1 ).

66 Gauss-Bonnet Theorem and Crofton formulas for complex planes
Proof. First, we relate the Gauss curvature integral with Γ 0,0 (Ω) from Example 2.4.20.1
M 2n−1 (∂Ω) = 2c−1 n,0,0
(n − 1)! Γ 0,0(Ω) = 2nω 2n Γ 0,0 (Ω). (4.6)
Thus, from Proposition 4.1.7 and using that c −1
n,1,0 = (n − 1)!ω 2n−1, c −1
n,3,1 = (n − 2)!ω 2n−3 and
ω 2n−1 = (2n − 1)ω 2n−3 /2π we obtain the result:
δ X M 2n−1 (∂Ω) =
2ɛ
(n − 1)! (−c−1 n,1,0 (3n − 1) ˜B 1,0 + 2c −1
n,1,0 (n − 1)˜Γ 1,0 + c −1
n,3,1 3ɛ(n − 1) ˜B 3,1 )
= 2ɛ(−ω 2n−1 (3n − 1) ˜B 1,0 + 2ω 2n−1 (n − 1)˜Γ 1,0 + 3ɛω 2n−3 ˜B3,1 )
= 2ɛω 2n−1 (−(3n − 1) ˜B 1,0 + 2(n − 1)˜Γ 1,0 + 3
2π ɛ(2n − 1) ˜B 3,1 ).
4.2 Variation of the measure of complex r-planes intersecting
a regular domain
Proposition 4.2.1. Let Ω ⊂ CK n (ɛ) be a regular domain, X a smooth vector field on CK n (ɛ)
with flow φ t and Ω t = φ t (Ω). Then
∫
∫
( ∫ )
d
dt∣ χ(Ω t ∩ L r )dL r = 〈∂φ/∂t, N〉 σ 2r (II| V )dV dx
t=0 G C n,r (Dp)
L C r
∂Ω
where N is the outward normal field, D is the tangent distribution to ∂Ω and orthogonal to
JN and σ 2r (II| V ) denotes the 2r-th symmetric elementary function of II restricted to V ∈
G C n−1,r (D p).
Proof. The proof is similar to the one in [Sol06, Theorem 4] for real space forms.
Denote G C n−1,r (D) = {(p, l) | l ⊆ T p∂Ω, dim R l = 2r and Jl = l} = ⋃ p∈∂Ω GC n−1,r (D p).
For each V ∈ G C n−1,r (D p), we take the parallel translation V t of V along φ t (x). Recall that
parallel translation preserves the complex structure (cf. [O’N83, page 326]). Then we project
V t orthogonally to D φt(x), obtaining a complex r-plane V ′
t (at least for small values of t). We
define
γ : G C n−1,r (D) × (−ɛ, ɛ) −→ LC r
((x, V ), t) ↦→ exp φt(x) V t ′ .
Proposition 3 in [Sol06] remains true, without change, in complex space forms. From this
proposition and using a similar argument as in [Sol06, teorema 4] we get
∫
∫
d
1
∑ ∂φ
dt∣ χ(Ω t ∩ L r )dL r = lim
sign〈
t=0 L C h→0 h
r
G C n−1,r (D)×(0,h) ∂t , N〉 sign(σ 2r(II| V ))γ ∗ dL r
∫
= 〈 ∂φ
G C n−1,r (D) ∂t , N〉 sign(σ 2r(II| V ))γ0(ι ∗ dφ∂t dL r )
where the sum on the second integral runs over the tangencies of L r with the hypersurfaces
∂Ω t with 0 < t < h.
Consider a J-moving frame {g; g 1 , Jg 1 , ..., g n , Jg n } such that g((p, l), t) = φ(p, t), γ =
〈g; g 1 , Jg 1 , ..., g r , Jg r 〉 ∩ CK n (ɛ) and Jg n ((p, l), t) = N t (outward unit vector to ∂Ω t at φ(p, t)).
We may assume that the moving frame is defined in a neighborhood of L C r since we are only
interested in regular points of γ.
(4.7)

4.2 Variation of the measure of complex r-planes intersecting a domain 67
Considerem the curve L r (t) given by the parallel translation of L r along the geodesic
given by N, the outward normal vector to ∂Ω 0 . Recall that parallel translations preserves the
complex structure (cf. [O’N83, page 326]). If P ∈ T Lr L C r denotes the tangent vector to L r (t)
at t = 0, then
ω i (P ) = 〈dg(P ), g i 〉 = 〈 d dt g(L r(t)), g i 〉 = 0, i ∈ {r + 1, r + 1, ..., n − 1, n − 1},
ω n (P ) = 〈dg(P ), N〉 = 1, (4.8)
ω ij (P ) = 〈∇g i (P ), g j 〉 = 〈 D dt g i(L r (t)), g j 〉 = 0,
j ∈ {r + 1, r + 1, ..., n, n}, i ∈ {1, 1, ..., r, r}.
The measure of complex r-planes in CK n (ɛ) is (cf. Proposition 1.5.5)
From (4.8), we get
dL r =
∣
n∧
i=r+1
ω i ∧ ω i
∧
i=1,...,r
j=r+1,...,n
dL r = |ω n |ι P dL r
since ι P dL r = | ∧ n−1
h=r+1 ω h ∧ ω h ∧ ω n
∧
ωij |.Thus,
with
and we get
ω ij ω ij
∣ ∣∣∣∣∣∣∣
.
ι dγ∂t dL r = |ω n (dγ∂t)|ι P dL r + |ω n |ι dγ∂t ι P dL r
ω n (dγ∂t) = 〈dg(dγ∂t), N〉 = 〈 ∂φ
∂t , N〉,
γ ∗ 0(ω n )(v) = 〈dg(dγ 0 (v)), N〉 = 0 ∀v ∈ T (p,l) G C n−1,r(T p ∂Ω 0 ),
γ ∗ 0(ι dγ∂t dL r ) = |〈 ∂φ
∂t , N〉|γ∗ 0(ι P dL r ).
Finally, using that ψ ∗ 0 (ι P dL r ) = |σ 2r (II| V )|dV dx, we get the result.
Remark 4.2.2. The integral
∫
G C n,r
σ 2r (II| V )dV (4.9)
seems difficult to compute directly. However, we will find it by an indirect method. Recall that
the analogous integral in real space forms is a multiple of an elementary symmetric function
of the principal curvatures.
For r = n − 1, the integral (4.9) can be easily computed in CK n (ɛ).
Corollary 4.2.3. Let Ω ⊂ CK n (ɛ) be a regular domain, X a smooth vector field on CK n (ɛ)
with flow φ t , and Ω t = φ t (Ω). Then,
∫
d
dt∣ t=0
L C n−1
χ(L n−1 ∩ Ω t )dL n−1 = ω 2n−1 ˜B1,0 (Ω).

68 Gauss-Bonnet Theorem and Crofton formulas for complex planes
Proof. From Proposition 4.2.1 we get the result since there is just one complex hyperplane
tangent to a point in ∂Ω. Thus,
∫
∫
∫
d
dt∣ χ(Ω t ∩ L n−1 )dL n−1 = 〈∂φ/∂t, N〉 σ 2n−2 (II| V )dV dx
t=0 L C n−1
∂Ω 0 G C n−1,n−1
∫
= 〈∂φ/∂t, N〉σ 2n−2 (II| D )dx
∂Ω 0
= 〈∂φ/∂t, N〉
∫∂Ω β ∧ θn−1 0
0
(n − 1)!
= c−1 n,1,0
(n − 1)! ˜B 1,0 (Ω)
= ω 2n−1 ˜B1,0 (Ω).
4.3 Measure of complex r-planes meeting a regular domain
4.3.1 In the standard Hermitian space
Using that the measure of complex r-planes in C n meeting a regular domain is a linear combination
of the Hermitian intrinsic volumes, and Propositions 4.2.1 and 4.1.7 we find explicitly
the coefficients of this linear combination.
Theorem 4.3.1. Let Ω ⊂ C n be a convex domain, X a smooth vector field over C n , φ t the
flow associated to X and Ω t = φ t (Ω). Then
∫
( )
d
n − 1 −1 ( ) n
dt∣ χ(Ω t ∩ L r )dL r = vol(G C n−1,r)ω 2r+1 (r + 1)
−1·
t=0 L C r r
r
⎛
⎞
n−r−1
∑
( )
· ⎝
2n − 2r − 2q − 1 1
n − r − q 4 ˜B n−r−q−1 2n−2r−1,q (Ω) ⎠ , (4.10)
and
∫
L C r
q=max{0,n−2r−1}
( ) n − 1 −1 ( n
χ(Ω ∩ L r )dL r = vol(G C n−1,r)ω 2r
r r
⎛
· ⎝
n−r
∑
q=max{0,n−2r}
) −1·
( )
1 2n − 2r − 2q
4 n−r−q n − r − q
Proof. In order to simplify the following computations, we consider
and
B k,q ′ = B′ k,q (Ω) := c−1
n,k,q µ k,q(Ω),
˜B ′ k,q = c−1 n,k,q ˜B k,q ,
⎞
µ 2n−2r,q (Ω) ⎠ . (4.11)
Γ ′ 2q,q = Γ ′ 2q,q(Ω) := 2c −1
n,2q,q µ 2q,q(Ω). (4.12)
˜Γ′ k,q = 2c −1
n,k,q ˜Γ k,q . (4.13)
The functional ∫ L C r χ(Ω ∩ L r)dL r is a valuation on C n with degree of homogeneity 2n −
2r. Thus, it can be expressed as a linear combination of the elements of degree 2n − 2r,
{µ 2n−2r,q | max{0, n − 2r} ≤ q ≤ n − r} (cf. Definition 2.4.11). Then, by Remark 2.4.12 and
(2.8), we have
∫
L C r
χ(Ω ∩ L r )dL r =
n−r−1
∑
q=max{0,n−2r}
C q B ′ 2n−2r,q + DΓ ′ 2n−2r,n−r (4.14)

4.3 Measure of complex r-planes meeting a regular domain 69
for certain constants C q , D which we wish to determine. This will be done by comparing the
variation of both sides of this equality.
From here on we assume 2r < n. The case 2r ≥ n can be treated in the same way (cf.
Remark 4.3.2).
By Proposition 4.1.7, the variation of the right hand side of (4.14) is a linear combination
of the following type
n−r−1
∑
q=n−2r−1
c q ˜B′ 2n−2r−1,q +
n−r−1
∑
q=n−2r
d q ˜Γ′ 2n−2r−1,q (4.15)
where the coefficients c q and d q can be expressed in terms of a linear combination with known
coefficients of the variables C q and D, that still remain unknown.
The variation of the left hand side of (4.14), by Proposition 4.2.1 is
∫
∫
∫
d
dt∣ χ(Ω t ∩ L r )dL r = 〈∂φ/∂t, N〉 σ 2r (II| V )dV dx. (4.16)
t=0
L C r
∂Ω
G C n−1,r
From Lemma 2.4.18 when pulling-back the form γ k,q from N(Ω) to ∂Ω, one gets a polynomial
expression P k,q of degree 2n − k − 1 in the coefficients h ij of II with i, j ∈ {1, 2, 2, . . . , n, n}.
Moreover, for each q the monomials in P k,q containing only entries of the form h ii contain the
factor h 11 = II(JN, JN) and do not appear in any other P k,q ′ with q ′ ≠ q. Therefore, every
non-trivial linear combination of {P k,q } q must contain the variable h 11 . On the other hand,
the integral ∫ G
σ C 2r (II| V )dV is a polynomial of the second fundamental form II restricted
n−1,r
to the distribution D = 〈N, JN〉 ⊥ , hence a polynomial not involving h 11 . Comparing the
expressions of (4.15) and (4.16), it follows that d q = 0 for all q ∈ {n − 2r, . . . , n − r − 1}.
As c q and d q depend on C q and D, we will obtain the value of c q once we know the value
of C q and D. We will get their value from the equalities {d q = 0}. Note that this gives r
equations, since q runs from n − 2r to n − r − 1 in (4.15). As for the unknowns, we need to
find r constants C q plus the constant D in (4.14).
We will get an extra equation by taking II| D = Id and equating (4.16) to (4.15). Then,
for any pair (n, r) we have a compatible linear system since constants in (4.14) exist. Next we
find the solution.
Let us relate explicitly the coefficients {c q } and {d q } in (4.15) with C q and D in (4.14).
To simplify the range of the subscripts, we denote d n−r−a with a = 1, . . . , r and c n−r−a with
a = 1, . . . , r + 1.
Coefficient d n−r−1 . From the variation of B ′ k,q in Cn (Proposition 4.1.7), the coefficient of
˜Γ ′ 2n−2r−1,n−r−1 comes from the variation of B′ 2n−2r,n−r−1 and Γ′ 2n−2r,n−r . Then,
d n−r−1 = −2r(n − r)D + (2n − 2r − 2(n − r − 1)) 2 C n−r−1
= 4C n−r−1 − 2r(n − r)D. (4.17)
Coefficient d n−r−a , a = 2, . . . , r. The coefficient of ˜Γ ′ 2n−2r−1,n−r−a comes from the variation
of B 2n−2r,n−r−a ′ and B′ 2n−2r,n−r−a+1 . Then,
d n−r−a = (2n − 2r − 2(n − r − a)) 2 C n−r−a − (2r + n − r − a + 1 − n)(n − r − a + 1)C n−r−a+1
= 4a 2 C n−r−a − (r − a + 1)(n − r − a + 1)C n−r−a+1 . (4.18)
Coefficient c n−r−1 . The coefficient of ˜B ′ 2n−2r−1,n−r−1 comes from the variation of B′ 2n−2r,n−r−1
and Γ ′ 2n−2r,n−r . Then,
c n−r−1 = 4(r + 1/2)(n − r)D − 4C n−r−1
= 2(2r + 1)(n − r)D − 4C n−r−1 . (4.19)

70 Gauss-Bonnet Theorem and Crofton formulas for complex planes
Coefficient c n−r−a , a = 2, . . . , r − 2. The coefficient of ˜B′ 2n−2r−1,n−r−a comes from the
variation of B ′ 2n−2r,n−r−a and B′ 2n−2r,n−r−a+1 . Then,
c n−r−a = −2(2a)(2a − 1)C n−r−a + 2(r − a + 3/2)(n − r − a + 1)C n−r−a+1
= −4a(2a − 1)C n−r−a + (2r − 2a + 3)(n − r − a + 1)C n−r−a+1 . (4.20)
Coefficient c n−2r−1 . The coefficient of ˜B ′ 2n−2r−1,n−2r−1 comes from the variation of B′ 2n−2r,n−2r .
Then,
c n−2r−1 = (2r − 2(r + 1) + 3)(n − r − (r + 1) + 1)C n−2r
= (n − 2r)C n−2r . (4.21)
Now, we solve the linear system given by {d n−r−a = 0} where a ∈ {1, . . . , r}. From
equations (4.17) and (4.18) the system is given by
{ r(n − r)D = 2Cn−r−1
4a 2 C n−r−a = (n − r − a + 1)(r − a + 1)C n−r−a+1 .
Thus,
(n − r − a + 1) · · · · · (n − r)(r − a + 1) · · · · r
C n−r−a =
2 · 4 a−1 a 2 (a − 1) 2 · · · · · 1 2 D
(n − r)!r!
=
2 2a−1 (n − r − a)!(r − a)!a!a! D
= D ( )( n − r r
2 2a−1 . (4.22)
a a)
To obtain the value of D, we compute ∫ G
σ C 2r (p)dV and β 2n−2r−1,n−r−a ′ in case II| D(p) =
n−1,r
λId for λ > 0, which occurs when Ω is a geodesic ball. On one hand, we have
∫
σ 2r (p)(λId| V )dV = λ 2r vol(G C n−1,r).
G C n−1,r
On the other hand, if II| D = λId, then the connection forms satisfy α 1i = λω i and β 1i = λω i .
Thus, θ 1 = 2λθ 2 and θ 0 = λ 2 θ 2 and we obtain
β ′ 2n−2r−1,n−r−a(p) = λ 2r (β ∧ θ r−a+1
0 ∧ θ 2a−2
1 ∧ θ n−r−a
2 )(p)
= 2 2a−2 λ 2r (β ∧ θ n−1
2 )(p) = 2 2a−2 λ 2r (n − 1)!.
So, the equation
∑r+1
vol(G C n−1,r) = c n−r−a 2 2a−2 (n − 1)!
a=1
must be satisfied.

4.3 Measure of complex r-planes meeting a regular domain 71
Substituting equations (4.19), (4.20) and (4.21) in the last equation gives
vol(G C n−1,r )
(n − 1)!
= (2(2r + 1)(n − r)D − 4C n−r−1 )
r∑
+ 2 2a−2 ((2r − 2a + 3)(n − r + a + 1)C n−r−a+1 − 4a(2a − 1)C n−r−a )
a=2
Thus,
+ 2 2r (n − 2r)C n−2r
= 2(2r + 1)(n − r)D + 4((2r − 1)(n − r − 1) − 1)C n−r−1
∑r−1
+ (−2 2a−2 4a(2a − 1) + 2 2a (2r − 2a + 1)(n − r − a))C n−r−a
a=2
+ (2 2r (n − 2r) − 2 2r−2 4r(2r − 1))C n−2r
r∑
= 2(2r + 1)(n − r)D + 2 2a ((2r − 2a + 1)(n − r − a) − a(2a − 1))C n−r−a
(
(4.22)
= D 2(n − r)!r!
2 n!
= D
r!(n − r − 1)!
r∑
a=0
a=1
D = vol(GC n−1,r )
2 n!
C n−r−a = vol(GC n−1,r )
4 a n!
(2r − 2a + 1)(n − r − a) − a(2a − 1)
(n − r − a)!(r − a)!a!a!
( ) n − 1 −1
,
r
( ) n − 1 −1 ( )( n − r r
r a a)
and, for 2r < n, we have
∫
r∑
χ(Ω ∩ L r )dL r = C n−r−a B 2n−2r,n−r−a ′ + DΓ ′ 2n−2r,n−r
L C r
and
∫
d
dt∣ t=0
L C r
a=1
= vol(GC n−1,r )
2 n!
( ) (
n − 1
−1 r∑ ( n − r
r
a
a=1
χ(Ω t ∩ L r )dL r = (2(2r + 1)(n − r)D − 4C n−r−1 )B ′ 2n−2r−1,n−r−1
+
)
)( r
a)
2 −2a+1 B ′ 2n−2r,n−r−a + Γ ′ 2n−2r,n−r
r∑
((2r − 2a + 3)(n − r + a + 1)C n−r−a+1 − 4a(2a − 1)C n−r−a )B 2n−2r−1,n−r−a
′
a=2
+ (n − 2r)C n−2r B 2n−2r−1,n−2r−1
′
= vol(GC n−1,r ) ( ) (
n − 1
−1 ∑r+1
( )( ) )
n − r r + 1 a
n! r
a a 4 ˜B ′ a−1 2n−2r−1,n−r−a . (4.23)
a=1
Finally, we use the relation in (4.12) and (2.8) to obtain the result.
Remark 4.3.2. If 2r ≥ n, then formula (4.10) follows directly from the relations among the
different bases of valuations on C n given in [BF08] and the following relation in [Ale03]
∫
χ(Ω ∩ L r )dL r = 1 ∫
M 2r−1 (∂Ω ∩ L r )dL r = cU
O 2(n−r),n−r
2r−1
L C r
L C r
)

72 Gauss-Bonnet Theorem and Crofton formulas for complex planes
for a certain constant c coming from the different normalizations in dL r .
Corollary 4.3.3. Let Ω ⊂ CK n (ɛ) be a regular domain, X a smooth vector field over CK n (ɛ),
φ t the flow associated to X and Ω t = φ t (Ω). Then
∫
( )
d
n − 1 −1 ( ) n
dt∣ χ(Ω t ∩ L r )dL r = vol(G C n−1,r)ω 2r+1 (r + 1)
−1·
t=0 L C r r
r
⎛
⎞
n−r−1
∑
( )
· ⎝
2n − 2r − 2q − 1 1
n − r − q 4 ˜B n−r−q−1 2n−2r−1,q (Ω) ⎠ . (4.24)
q=max{0,n−2r−1}
Proof. Comparing equation (4.10) and Proposition 4.2.1 in case ɛ = 0 shows that if Ω is a
regular convex domain, then
∫
)
〈X, N〉 σ 2r (II|V )dV dx
∂Ω
( ∫
G C n,r
equals the right hand side of equation above. By taking a vector field X that vanishes outside
an arbitrarily small neighborhood of a fixed x ∈ ∂Ω, we deduce the following equality between
forms
(∫ )
σ 2r (II| V )dV dx = ω 2r+1
)( n
G C n−1,r r)vol(G C n−1,r)(r + 1)· (4.25)
(Tx∂Ω)
·
n−r−1
∑
q=max{0,n−2r−1}
( n−1
r
( 2n − 2r − 2q − 1
n − r − q
)
cn,2n−2r−1,n−r−q
4 n−r−q−1 β ∧ θ r−q+1
0 ∧ θ 2q−2
1 ∧ θ n−r−q
2 .
This equation can be written as P (II)dx = Q(II)dx where P and Q are polyomials with entries
in the second fundamental form. These polynomials concide for any positive defined bilineal
form. Thus, P = Q and (4.25) holds for regular domains (not necessarily convex domains).
Moreover, it is valid in CK n (ɛ) for any ɛ. Applying Proposition 4.2.1 we get the result.
Corollary 4.3.4. Equation (4.11) holds for any regular domain not necessarily convex.
Proof. Consider Ω t = φ t (Ω) with φ t a given flow.
From the last corollary, it is known the variation of the left hand side of (4.11).
By Proposition 4.1.7, the variation of the right hand side is a linear combination of
{B k,q , Γ k,q }. By Theorem 4.3.1 this linear combination coincides with the right hand side
of (4.24).
Thus, the variation of both sides of (4.11) coincides. So, the difference between both
members of (4.11) is constant.
Take φ t such that φ t (Ω) converges to a point for t → ∞. Both sides of (4.11) tend to zero
when t → ∞, thus their difference vanishes for all t.
4.3.2 In complex space forms
Theorem 4.3.5. Let Ω be a regular domain in CK n (ɛ). Then
∫
( ) n − 1
χ(Ω ∩ L r )dL r = vol(G C n−1,r)
−1· (4.26)
L C r
r
⎛
⎞
n−1
∑
( ) n −1 ∑k−1
( )
· ( ɛ k−(n−r) ω 2n−2k
⎝
1 2k − 2q
k
4 k−q µ 2k,q (Ω) + (k + r − n + 1)µ 2k,k (Ω) ⎠
k − q
k=n−r
+ ɛ r (r + 1)vol(Ω)).
q=max{0,2k−n}

4.3 Measure of complex r-planes meeting a regular domain 73
Proof. We will show that both sides have the same variation δ X with respect to any vector
field X. This implies the result: one can take a deformation Ω t of Ω such that Ω t converges
to a point. Then both sides of (4.26) have the same derivative, and both vanish in the limit.
The variation of the left hand side of (4.26) is given by Corollary 4.3.3. The variation of
the right hand side can be computed by using Proposition 4.1.7, and δ X V = 2 ˜B 2n−1,n−1 . In
order to simplify the computations we rewrite the right hand side of (4.26) as
+
n−1
∑
j=n−r
By Proposition 3.8 we have
+
n−1
∑
j=n−r
C r (Ω) := vol(GC n−1,r )
n!
⎛
ɛ j−n+r ⎝ j − n + r + 1 Γ ′ 2j,j +
2
δ X C r (Ω) = vol(GC n−1,r)
n!
( ) n − 1 −1
{ɛ r (r + 1)n!V
r
j−1
∑
q=max(0,2j−n)
( )
1 n − j j
4 j−q j − q)(
q
B ′ 2j,q
⎞
⎠}.
( ) −1 n − 1
[ɛ r n(r + 1)δ x ˜B′ 2n−1,n−1 (4.27)
r
ɛ j−n+r j − n + r + 1 {−2(n − j)j˜Γ ′ 2j−1,j−1 + 2ɛ(n − j − 1)(j + 1)˜Γ ′ 2j+1,j
2
+4(n − j + 1 (
2 )j ˜B j + 1
2j−1,j−1 ′ + 4ɛ − (n − j)(2j + 3 )
2
2 ) ˜B 2j+1,j ′ + 4ɛ 2 (n − j − 1)(j + 3 2 ) ˜B 2j+3,j+1}]
′
+
n−1
∑
∑j−1
j=n−r q=max{0,2j−n}
ɛ j−n+r ( ) n − j j
4 j−q {(2j − 2q)
j − q)( 2˜Γ′ q
2j−1,q
−(n + q − 2j)q˜Γ ′ 2j−1,q−1 + 2(n + q − 2j + 1 2 )q ˜B ′ 2j−1,q−1 − 2(2j − 2q)(2j − 2q − 1) ˜B ′ 2j−1,q
+2ɛ(2j − 2q)(2j − 2q − 1) ˜B ′ 2j+1,q+1 − 2ɛ(n − 2j + q)(q + 1 2 ) ˜B ′ 2j+1,q}.
We will show that the previous expression is independent of ɛ; i.e. all the terms containing ɛ
cancel out. Hence, δ X C r (Ω) coincides with (4.24) since we know this happens for ɛ = 0. This
will finish the proof.
We concentrate first on the terms with ˜B k,q ′ . By putting together similar terms, the third
line of (4.27) is
n−1
∑
h=n−r+1
ɛ h−n+r 2{(h − n + r + 1)(n − h + 1 2 )h + (h − n + r)(h 2 − (n − h + 1)(2h − 1 )) (4.28)
2
+(h − n + r + 1)(n − h + 1)(h − 1 2 )} ˜B ′ 2h−1,h−1
−ɛ r {(r + 2)n − 1} ˜B ′ 2n−1,n−1 + (2r + 1)(n − r) ˜B ′ 2n−2r−1,n−r−1.
By putting together similar terms, the double sum in (4.27) (forgetting for the moment
the terms with ˜Γ ′ k,q ) becomes
n−1
∑
h−2
∑
h=n−r a=max(−1,2h−n−1)
−
n−1
∑
h−1
∑
h=n−r a=max(0,2h−n)
ɛ h−n+r ( )( )
n − h h
4 h−a−1 2(n + a − 2h + 3 h − a − 1 a + 1
2 )(a + 1) ˜B 2h−1,a
′
ɛ h−n+r ( )
n − h h
4 h−a 2(2h − 2a)(2h − 2a − 1)
h − a)(
a
˜B 2h−1,a
′

74 Gauss-Bonnet Theorem and Crofton formulas for complex planes
−
+
n∑
h−1
∑
h=n−r+1 a=max(1,2h−n−1)
n∑
h−2
∑
h=n−r+1 a=max(0,2h−n−2)
ɛ h−n+r ( n − h + 1
4 h−a h − a
ɛ h−n+r ( n − h + 1
4 h−a−1 h − a − 1
)( ) h − 1
2(2h − 2a)(2h − 2a − 1)
a − 1
˜B 2h−1,a
′
)( h − 1
a
)
2(n − 2h + a + 2)(a + 1 2 ) ˜B ′ 2h−1,a.
Note that the terms with a = −1 or a = 2h − n − 2 vanish, if they occur. Then, one checks
that all the terms in the above expression cancel out except those with h = n − r, n, and those
with a = h − 1. Clearly the terms corresponding to h = n − r are independent of ɛ. The terms
with h = n sum up ɛ r (n − 1) ˜B 2n−1,n−1 ′ , and together with the similar term appearing in (4.28)
cancel out the first term in (4.27). Finally, the terms with a = h − 1 are cancelled with the
sum in (4.28).
With a similar but shorter analysis one checks that the multiples of ˜Γ ′ k,q
cancel out completely.
This shows that (4.27) is independent of ɛ, and finishes the proof.
Remark 4.3.6. The coefficients of µ k,q and vol in (4.26) were found by solving a linear system
of equations, which we write down in the appendix.
4.4 Gauss-Bonnet formula in CK n (ɛ)
Theorem 4.4.1. Let Ω be a regular domain in CK n (ɛ). Then
ω 2n χ(Ω) =(n + 1)ɛ n vol(Ω) + (4.29)
⎛
⎞
n−1
∑ (n − c)ω 2n−2c ɛ c
+
n ( ∑c−1
( )
) ⎝
1 2c − 2q
n−1
4 c−q µ 2c,q (Ω) + (c + 1)µ 2c,c (Ω) ⎠ .
c − q
c
c=0
q=max{0,2c−n}
Remark 4.4.2. For ɛ = 0 we have the Gauss-Bonnet formula in C n ∼ = R 2n , where it is known
χ(Ω) = 1
2nω 2n
M 2n−1 (∂Ω) = µ 0,0 (Ω),
which coincides with the expression in the previous result.
Here we prove the certainty of (4.29) but in the appendix we give a constructive proof of
the result.
Proof. We proceed analogously to the proof of Theorem 4.3.5. In fact, the same computations
of the previous proof show (in case r = n) that the right hand side of (4.29) has null variation.
For ɛ = 0 equation (4.29) is the well know Gauss-Bonnet formula in C n ∼ = R 2n . For ɛ ≠ 0,
we take a smooth deformation of Ω to get a domain contained in a ball of radius r. Under
this deformation, the right hand side of (4.29) remains constant. By taking r small enough,
the difference between both sides can be made arbitarily small. Hence, they coincide.
Although in (4.29) does not appear the Gauss curvature, we can easily get the following
expression.
Corollary 4.4.3. Let Ω ⊂ CK n (ɛ) be a regular domain. Then,
2nω 2n χ(Ω) =M 2n−1 (∂Ω) + 2n(n + 1)ɛ n vol(Ω)+
⎛
n−1
∑
( ) n −1
+ 2nω 2n−2c ɛ c ⎝
c
c=1
∑c−1
q=max{0,2c−n}
⎞
( )
1 2c − 2q
4 c−q µ 2c,q (Ω) + (c + 1)µ 2c,c (Ω) ⎠ .
c − q

4.4 Gauss-Bonnet formula in CK n (ɛ) 75
Proof. Apply relation (4.6) in (4.29).
Remarks 4.4.4. 1. The Gauss-Bonnet-Chern formula in spaces of constant sectional curvature
k and even dimension, for a regular domain Ω is given by
O 2m−1 χ(Ω) = M 2n−1 (∂Ω) + c n−3 M 2n−3 (∂Ω) + · · · + c 1 M 1 (∂Ω) + (|k|) n/2 vol(Ω),
where c j are known and depend on the curvature k.
Note that in the previous expression appear all mean curvature integrals with odd subscript
and the volume. In formula (4.29) in CK n (ɛ), ɛ ≠ 0, also appear all the Hermitian
intrinsic volumes in CK n (ɛ) with the first subscript odd.
2. In [Sol06] it is given an expression of the Gauss-Bonnet-Chern formula in space of constant
sectional curvature k using the measure of planes of codimension 2 meeting the
domain. The obtained formula for Ω ⊂ RK n (ɛ) is
nω n χ(Ω) = M n−1 (∂Ω) +
2k ∫
χ(Ω ∩ L n−2 )dL n−2 . (4.30)
ω n−1 L n−2
A natural question is whether in complex space forms, there exists a similar expression
relating the Gauss curvature integral with the Euler characteristic and the measure of
some complex planes meeting the domain
or
c 0 χ(Ω) = n−1
∑
∫
M 2n−1 (∂Ω) + c q
q=1
c 0 χ(Ω) = n−1
∑
n−1
∑
∫
M 2n−1 (∂Ω) + M 2q+1 (∂Ω) + c q
q=0
L C q
χ(Ω ∩ L q )dL q
q=1
L C q
χ(Ω ∩ L q )dL q .
Taking variation on both sides in these expressions, we get that these expressions cannot
hold in general (for n = 2 and n = 3 we can choose constants satisfying them). Anyway,
in Theorem 4.4.5 we give a similar expression. Perhaps, if we knew a formula for the
measure of totally real planes meeting a domain we could find a more similar expression.
3. For n = 2, the Gauss-Bonnet-Chern formula was already known in CK n (ɛ). It was given
in [Par02]. From Theorem 4.4.1 we get the same expression, which can be written as
χ(Ω) = 1 ( ( )
)
1 1
π 2 2 Γ′ 0,0 + ɛ
4 B′ 2,0 + Γ ′ 2,1 + 6ɛ 2 vol(Ω) .
This expression can also be stated as
χ(Ω) = 1 (
2π 2 M 3 (∂Ω) + 3ɛ ( ∫
M 1 (∂Ω) +
2
∂Ω
)
)
k n (JN) + 12ɛ 2 vol(Ω)
(4.31)
and
(
χ(Ω) = 1
∫
2π 2 M 3 (∂Ω) + 2ɛ χ(∂Ω ∩ L 1 )dL 1 + ɛ ∫
)
k n (JN) + 12ɛ 2 vol(Ω) .
L C 2
1
∂Ω
In the following result we express the Euler characteristic in terms of the Gauss curvature
integral, the volume, the measure of complex hyperplanes meeting a domain and the valuations
µ 2c,c . This formula generalizes (4.30) in complex space forms.

76 Gauss-Bonnet Theorem and Crofton formulas for complex planes
Theorem 4.4.5. Let Ω ⊂ CK n (ɛ) be a regular domain CK n (ɛ). Then,
∫
ω 2n χ(Ω) = ɛ χ(Ω ∩ L n−1 )dL n−1 +
L C n−1
= 1
∫
2n M 2n−1(∂Ω) + ɛ
L C n−1
n∑
c=0
ɛ c ω 2n
µ 2c,c (Ω)
ω 2c
χ(Ω ∩ L n−1 )dL n−1 +
n∑
c=1
ɛ c ω 2n
µ 2c,c (Ω).
ω 2c
Proof. From Theorems 4.3.5 and 4.4.1 we get the stated formula
⎛
⎞
n−1
∑ ɛ c n−1
c! ∑
( )
χ(Ω) = ⎝
1 2c − 2q
π c
4 c−q B 2c,q (Ω) + (c + 1)Γ 2c,c (Ω) ⎠ + ɛn (n + 1)!
c − q
π n vol(Ω)
c=0 q=max{0,2c−n}
⎛
⎞
n−1
∑ ɛ c n−1
c! ∑
( )
= Γ 0,0 (Ω)+ ⎝
1 2c − 2q
π c 4 c−q B 2c,c (Ω) + (c + 1)Γ 2c,c (Ω) ⎠+ ɛn (n + 1)!
c − q
π n vol(Ω)
c=1 q=max{0,2c−n}
⎛
⎞
= Γ 0,0 (Ω) + ɛ n! n−1
∑ ɛ c−1 c!π n−c n−1
∑
( )
⎝
1 2c − 2q
π n n!
4 c−q B 2c,c (Ω) + cΓ 2c,c (Ω) ⎠ +
c − q
+ ɛ n! n−1
∑
π n
c=1
c=1
= Γ 0,0 (Ω) + ɛ n!
π n ∫
ɛ c−1 c!π n−c
n!
L C n−1
q=max{0,2c−n}
Γ 2c,c (Ω) + ɛn (n + 1)!
π n vol(Ω)
n−1
∑
χ(Ω ∩ L n−1 )dL n−1 +
c=1
ɛ c (
c!
ɛ n )
π c Γ (n + 1)!
2c,c(Ω) +
π n − ɛn n!n
π n vol(Ω).
4.5 Another method to compute the measure of complex lines
meeting a regular domain
From Theorem 4.3.5 we can give an expression of the measure of complex lines meeting a
regular domain (just taking r = 1). Here, we give another method to obtain this expression,
using the results in Chapter 3.
4.5.1 Measure of complex lines meeting a regular domain in C n
Proposition 4.5.1. Let Ω ⊂ C n be a regular domain. Then,
∫
χ(Ω ∩ L 1 )dL 1 =
ω (
∫ )
2n−4
(2n − 1)M 1 (∂Ω) + k n (Jn) .
4n(n − 1)
∂Ω
L C 1
Proof. Recall that each complex line is isometric to C. Gauss-Bonnet formula in C n for
hypersurfaces ∂Ω states
M 2n−1 (∂Ω) = 2nω 2n χ(Ω).
Applying Gauss-Bonnet formula in C and Proposition 3.3.2 with s = 1 we get the result
∫
χ(Ω ∩ L 1 )dL 1 = 1 ∫ ∫
k g dpdL 1 = ω (
∫ )
2n−2
(2n − 1)M 1 (∂Ω) + k n (Jn) .
L C 2π
1
L C 1 ∂Ω∩L 1
4nω 2 ∂Ω

4.5 Another method to compute the measure of complex lines 77
Although Gauss-Bonnet formula is known in C n for n ≥ 1, we cannot apply the same
method to give the expression of the measure of s-planes meeting a regular domain since
the integral ∫ L C r M 2r−1(∂Ω ∩ L r )dL r is not in general known. In the next section we get
an expression for this integral using the Gauss-Bonnet formula and the measure of complex
r-planes meeting a regular domain.
4.5.2 Measure of complex lines meeting a regular domain in CP n and CH n
The following result is given, for instance, in [ÁPF04].
Proposition 4.5.2 ([ÁPF04]). Let Ω be a regular domain in CPn or CH n . Then,
∫
vol 2s (Ω ∩ L s )dL s = Cvol 2n (Ω).
L C s
The value of the constant C, it is not known, but now we shall need it explicitly.
Proposition 4.5.3. Let Ω be a regular domain in CP n or CH n . Then,
∫
vol 2s (Ω ∩ L s )dL s = vol(G C n,n−s)vol 2n (Ω).
L C s
Proof. In order to find C we apply last proposition to a ball of radius R. Let L s be a complex
s-plane meeting B R at a distance ρ from the center of the ball. From Lemma 3.2.13 in [Gol99],
we have that the intersection B R ∩ L s is a ball of complex dimension s and radius r such that
cos ɛ (R) = cos ɛ (r) cos ɛ (ρ).
The expression of the volume of a geodesic ball of radius R in CK n (ɛ) is (cf. [Gra73])
vol 2n (B R ) =
πn
|ɛ| n n! sin2n ɛ (R).
Using this expression we get
vol 2s (L s ∩ B R ) =
( ) π s
1
|ɛ| s!
( cos
2
ɛ (R) − cos 2 ) s
ɛ(ρ)
cos 2 .
ɛ(ρ)
On the other hand, the Jacobian of the change of variables to polar coordinates is given by
(cf. [Gra73])
cos ɛ (R) sinɛ
2n−1 (R)
|ɛ| n−1/2 .
Then, using Proposition 1.5.8, we get
∫
L C s
vol 2s (L s ∩ B R )dL s = πs vol(G C n,n−s)O 2(n−s)−1
|ɛ| n−1/2 s!
·
∫ R
0
s∑
( ) s
(−1) i+1 ·
i
i=0
sin 2(n−s)−1
ɛ (ρ) cos 2i
ɛ (R) cos 2(s−i)+1
ɛ (ρ)dρ
= πs vol(G C n,n−s)O 2(n−s)−1
|ɛ| n−1/2 ·
s!
s∑ ∑s−i
i∑
( )( )( 2(n−s+k+j)
s s − i i sin
· (−1) i+1 ɛ (R)
√ .
i j k)
ɛ(n − s + j)
i=0 j=0 k=0

78 Gauss-Bonnet Theorem and Crofton formulas for complex planes
From Proposition 4.5.2, this expression is a multiple of
vol 2n (B R ) =
πn
|ɛ| n n! sin2n ɛ (R).
Thus, all terms in the sum are zero except for 2(n − s + k + j) = 2n, i.e. k + j = s, which
together with j ≤ s − i, k ≤ i implies j = s − i, k = i, and we get
∫
vol 2s (L s ∩ B R )dL s
L C s
Finally, from equality
= πs vol(G C n,n−s)O 2(n−s)−1
|ɛ| n s!
= πs vol(G C n,n−s)O 2(n−s)−1
2|ɛ| n
s∑
( ) s sin
(−1) i+1 2n
ɛ (R)
i 2(n − i)
i=0
sin 2n
ɛ (R)
s∑
i=0
= πs vol(G C n,n−s)O 2(n−s)−1 (n − s − 1)!
2|ɛ| n n!
and using O 2(n−s)−1 = 2(n − s)ω 2(n−s) = 2
(−1) i+1
i!(s − i)!(n − i)
sin 2n
ɛ (R).
C
πn
|ɛ| n n! = πs vol(G C n,n−s)O 2(n−s)−1 (n − s − 1)!
2|ɛ| n ,
n!
πn−s
(n−s−1)!
, we get the value of the constant C.
Corollary 4.5.4. Let Ω ⊂ CK n (ɛ) be a regular domain. Then,
∫
χ(Ω ∩ L 1 )dL 1 =
ω (
∫
)
2n−4
(2n − 1)M 1 (∂Ω)+ k n (JN)+8nɛvol(Ω) .
4n(n − 1)
∂Ω
L C 1
where k n (JN) denotes the normal curvature in the direction JN.
Proof. Using Gauss-Bonnet formula in H 2 (−4) we have (cf. [San04, page 309])
∫
χ(Ω ∩ L 1 )dL 1 = 1 ∫
M 1 (∂Ω ∩ L 1 )dL 1 − 2 ∫
vol(Ω ∩ L 1 )dL 1 ,
2π L C π
1
L C 1
and using Proposition 4.5.3 with s = 1, and Proposition 3.3.2 we get the result.
Corollary 4.5.5. If Ω is a regular domain in CK 2 (ɛ), ɛ ≠ 0, then
∫
Ω∩L 1 ≠∅
χ(∂Ω ∩ L 1 )dL 1 = 1 4
(M 1 (∂Ω) − 1 3ɛ M 3(∂Ω) + 4ɛvol(Ω) + 2π2
3ɛ χ(Ω) )
.
Proof. From previous corollary, with n = 2, we have
∫
χ(Ω ∩ L 1 )dL 1 = 1 (
∫
)
3M 1 (∂Ω) + k n (JN) + 16ɛvol(Ω) .
L C 8
1 ∂Ω
Isolating ∫ ∂Ω k n(JN) in expression (4.31) we get the stated result.
Note that the previous corollary cannot be extended to ɛ = 0 since the expression (4.31)
does not contain the term ∫ ∂Ω k n(JN).
L C 1

4.6 Total Gauss curvature integral C n 79
4.6 Total Gauss curvature integral C n
Theorem 4.6.1. If Ω ⊂ C n is a regular domain, then
∫
L C r
( ) n − 1 −1 ( ) n
M 2r−1 (∂Ω∩L r )dL r = 2rω2rvol(G 2 C n−1,r)
−1·
r r
⎛
⎞
n−r
∑
( )
· ⎝
1 2n − 2r − 2q
4 n−r−q µ 2n−2r,q (Ω) ⎠ .
n − r − q
q=max{0,n−2r}
Proof. On one hand, by Gauss-Bonnet formula in C n and the relation (4.6), we have
∫
∫
∫
M 2r−1 (∂Ω ∩ L r )dL r = 2rω r χ(Ω ∩ L r )dL r = 2rω 2r µ 0,0 (Ω ∩ L r )dL r . (4.32)
L C r
On the other hand, by Theorem 4.3.1, we have
∫
L C r
L C r
( ) n − 1 −1 ( n −1
χ(∂Ω ∩ L r )dL r = vol(G C n−1,r)ω 2r
r r)
⎛
⎞
n−r
∑
( )
· ⎝
1 2n − 2r − 2q
4 n−r−q µ 2n−2r,q (Ω) ⎠ .
n − r − q
q=max{0,n−2r}
If we equate both expressions and we use the relation (4.6), we get the result.
L C r

Chapter 5
Other Crofton formulas
In the previous chapter we give an expression for the measure of complex planes intersecting
a regular domain in a complex space form. Complex planes in CK n (ɛ) are totally geodesic
submanifolds, but, by Theorem 1.4.6, there are other totally geodesic submanifolds. Totally
real planes are also totally geodesic submanifolds in CK n (ɛ) for any ɛ (cf. Theorem 1.4.6).
Moreover, for ɛ = 0, all submanifolds generated by the exponential map of a vector subspace
holomorphically isometric to C k ⊕ R k−2p are totally geodesic. Note that complex planes and
totally real planes are particular cases of these submanifolds, for (k, p) = (2p, p) and (k, p) =
(k, 0), respectively.
In this chapter we obtain an expression for the measure of planes of type (2n−p, n−p), the
so-called coisotropic planes, intersecting a domain in C n , and an expression for the measure of
Lagrangian planes in CK n (ɛ).
5.1 Space of (k, p)-planes
First, we recall the definition of (k, p)-plane in C n , as it is given in [BF08].
Definition 5.1.1. Suppose that V is a real vector space and L n k
(V ) denote the space of all
affine subspaces of dimension k in V . If V = C n , considered as a real vector space, then the
space of (k, p)-planes, L k,p (C n ) ⊂ L n k (Cn ) is defined as the subset of all subspace of (real)
dimension k that can be expressed as the orthogonal direct sum of a complex subspace of
complex dimension p and a totally real subspace of (real) dimension (k − 2p).
We denote the elements of L k,p (C n ) by L k,p and the Grassmannian of all (k, p)-planes
through the origin in a vector space V by G n,k,p (V ).
From the previous definition, L k,p (C n ) is the orbit of C p ⊕ R k−2p under the action of
C n ⋊ U(n) (which are the holomorphic isometries of C n ).
The notion of (k, p)-plane was extended to CH n . In [Gol99] and [Hsi98], they are defined
as particular cases of the so-called linear submanifolds.
Definition 5.1.2. The image of the exponential map from a point x ∈ CH n of a vector
subspace in T x CH n is called linear submanifold.
The image of the exponential map from a point x ∈ CH n of a (k, p)-plane in T x CH n is
called linear (k, p)-plane.
This definition could be also stated in CP n , but doing so, we obtain submanifolds with
singularities except when the submanifold is totally geodesic, i.e. for complex planes (which
correspond to (2p, p)-planes), and for totally real planes (which correspond to (k, 0)-planes).
In CH n , linear submanifolds are not always totally geodesic submanifolds. They are totally
geodesic just for complex planes and totally real planes (cf. Theorem 1.4.6).
81

82 Other Crofton formulas
5.1.1 Bisectors
As in complex hyperbolic space there are no totally geodesic real hypersurfaces, it is natural
to look for real hypersurfaces with similar properties to those expected for a totally geodesic
real hypersurface. In [Gol99, page 152] it is answered that they are the so-called bisectors, also
denoted by spinal superfaces.
Definition 5.1.3. Let z 1 , z 2 be two different points in CH n . The bisector equidistant from z 1
and z 2 is defined as
E(z 1 , z 2 ) = {z ∈ CH n | d(z 1 , z) = d(z 2 , z)}
where d(z, z i ) denotes the distance between points z i and z at CK n (ɛ) (see Proposition 1.2.3).
Definition 5.1.4. Let z 1 , z 2 be two different points in CH n .
• The complex geodesic Σ defined by z 1 and z 2 is the complex spine of the bisector E(z 1 , z 2 ).
• The real spine of the bisector E(z 1 , z 2 ), σ(z 1 , z 2 ) is the intersection between the bisector
and the complex spine, i.e.
σ(z 1 , z 2 ) = E(z 1 , z 2 ) ∩ Σ(z 1 , z 2 ) = {z ∈ Σ | d(z 1 , z) = d(z 2 , z)}.
• A slide of E is a complex hyperplane Π −1
Σ (s) where Π : CHn −→ Σ denotes the orthogonal
projection over Σ.
Remark 5.1.5. 1. The set of all slides in a bisector defines a foliation of the bisector by
complex hyperplanes.
2. The real spine is a (real) geodesic in CH n since Σ is totally geodesic and isometric to
H 2 , and in the real hyperbolic space, the bisector line of two given points is a geodesic.
3. Each geodesic γ ⊂ CH n is the real spine of a unique bisector. Indeed, take the complex
line Σ containing γ and the orthogonal projection Π Σ to Σ. Then, Π −1
Σ
(γ) defines a
bisector.
Example 5.1.6. In CH 2 with the projective model, the bisector with respect to z 1 = [(1, 0, i)]
and z 2 = [(1, 0, −i)] is
E(z 1 , z 2 ) = {[(1, z, t)] ∈ CH n | z ∈ C, t ∈ R}.
This expression can be obtained directly using the formula for the distance between 2 points
given at Proposition 1.2.3. {[(
The complex spine is 1, 0, i λ − µ )]
}
with λ, µ ∈ C both nonzero and the real spine
λ + µ
is {[1, 0, t]}.
Proposition 5.1.7. The isometries of CH n act transitively over the space of bisectors.
Proof. Using the correspondence between bisectors and real geodesics, we have that isometries
act transitively over the space of bisectors, since they do so over the space of real geodesics.
It is known that there are no non-trivial isometries which fix pointwise a bisector , since
they are not totally geodesic hypersurfaces (if ɛ ≠ 0). Anyway, we can consider the reflection
with respect to a slice S of the bisector. This reflexion fixes pointwise the slice S and lies the
bisector invariant. Moreover, each of these reflexions is also a reflection with respect to the
spine σ, thus, it fixes the points in σ ∩ S.

5.1 Space of (k, p)-planes 83
Bisectors are hypersuperfaces of cohomogenity one. That is, the orbit, for the group of the
isometries fixing a bisector, of a point (except for the points in the real spine, which are of null
measure in the bisector) in a bisector is a submanifold of codimension 1 inside the bisector; a
submanifold of codimension 2 in CH n (cf. [GG00]).
Thus, bisectors are not homogeneous hypersurfaces. By definition a hypersurface is homogeneous
if the orbit of each point (for the isometry group fixing the hypersurface) is all the
hypersurface.
Let us study the orbit of a point in a bisector.
A geodesic γ(t) ⊂ CH n uniquely determines a tube T (r) of radius r and a bisector E with
real spine γ.
We study the relation between these two hypersurfaces, obtaining the orbit of a point in
the bisector in terms of the tube containing the point.
Proposition 5.1.8. Let p ∈ E. Consider r such that p ∈ T (r) ∩ E where T (r) denotes the
tube of radius r along the real spine γ of E. The orbit of p by the isometries fixing E, is given
by T (r) ∩ E.
Proof. Each point of the orbit O p of p belongs to T (r) since the isometries fixing the bisector
fix the spine, and they preserve distances. Then, O p ⊂ T (r) ∩ E.
Every point in T (r) ∩ E belongs to the orbit of p. Indeed, if q ∈ T (r) ∩ E then d(q, γ(t)) =
d(p, γ(t)), a necessary condition to be q in the orbit of p. The projection of the points p, q to
Σ can or cannot be the same point. Let us prove that in both cases there exists an isometry
g such that fixes the bisector and g(p) = q.
Suppose that p and q project at Σ to the same point x. Then p and q belong to the
same slide of E. Let us define an isometry g fixing x and the bisector. We denote by v the
tangent vector to the real spine γ at x. As the isometries fixing the bisector also fix γ, g
satisfies dg(v) = ±v. Moreover, as isometries preserve the holomorphic angle dg(Jv) = ±Jv.
Thus, g fixes the complex spine Σ and its orthogonal complement at x, which is the slide
containing p and q, and is isometric to CH n−1 . Now, in CH n−1 there exists an isometry ˜g
such that ˜g(p) = q (since CH n−1 is a homogeneous space). Therefore, g defined by g(x) = x,
dg(v) = ±v, dg(Jv) = ±Jv and dg(u) = ˜dg(u), for all u ∈ 〈v, Jv〉 ⊥ , gives an isometry of
CK n (ɛ) fixing E and such that g(p) = q.
Suppose that p and q do not project at Σ to the same point. Let x = Π Σ p and y = Π Σ q.
Note that x, y ∈ γ since p and q are points in the bisector. Then, there exists a reflection ρ
such that ρ(x) = y and ρ(γ) = γ. Thus, dρ takes the orthogonal space of {γ x, ′ Jγ x} ′ to the
orthogonal space of {γ y, ′ Jγ y}. ′ Moreover, ˜q = ρ(q) satisfies Π Σ˜q = Π Σ q. If we consider the
points ˜q and q, then we are in the previous case and we know that there exists an isometry g
such that g(˜q) = q.
From this proposition we have that the subset of bisectors containing a point is a noncompact
set, in the space of bisectors. In the next proposition, we prove that the measure of
bisectors meeting a regular domain is infinite.
Remark 5.1.9. Denote by dL the invariant density of the space of bisectors B and by dL 1
the invariant density of the space of real geodesics in CH n . By the correspondence between
geodesics and bisectors we have
dL = dL 1 .
If Σ denotes the complex line containing a real geodesic γ, then the density of the space
of real geodesics can be expressed as
dL 1 = dL Σ 1 dΣ,

84 Other Crofton formulas
where dΣ denotes the invariant density of complex lines and dL Σ 1 the invariant density of the
space of real geodesics contained in Σ. Using polar coordinates ρ, θ in Σ we get
dL Σ 1 dΣ = cos 4ɛ (ρ)dρdθdΣ. (5.1)
Proposition 5.1.10. The measure of bisectors meeting a regular domain in CK n (ɛ) with ɛ < 0
is infinite.
Proof. We prove that the measure of bisectors intersecting a ball B of radius R in CK n (ɛ) is
infinite, i.e.
∫
χ(B ∩ L)dL = +∞.
B
Consider the expression for the density of bisectors in (5.1). Denote the volume element
of CK n (ɛ) by dx. Fixed a bisector L by x, then dx can be expressed as dx = dx 1 dx L where
dx L denotes the volume element in the bisector and dx 1 the length element in the direction
N x orthogonal to the bisector at x.
If N y is the normal vector to the bisector at y = Π Σ (x), then the plane spanned by N y
(which coincides with the normal vector to the real spine inside Σ) and the tangent vector u
to the geodesic joining y and x is a totally real plane and contains N x .
Thus, the plane exp y (span{N y , u}) is isometric to H 2 (ɛ). If r denotes the distance between
y and x, then dx 1 = cos ɛ (r)dy 1 where dy 1 denotes the length element in the direction N y .
In the previous remark, we give an expression for dL 1 . Now, we use it taking polar
coordinates with respect to y ∈ Σ, so ρ = 0. Then, dy 1 = dρ and
dx L dL 1 = dx L dL Σ 1 dΣ = dx L dθdρdΣ = dθdx L dy 1 dΣ =
1
cos ɛ (r) dθdΣdx.
On the other hand, fixed a regular domain Ω ⊂ CK n (ɛ) it follows, for some constant C > 0,
vol(Ω) < 1 C χ(Ω).
Then,
∫
∫
∫ ∫
χ(B ∩ L)dL > C vol(B ∩ L)dL = C dx L dL
B
∫
= C
B
B
∫
(1.16)
= 2πC
L C 1
∫
∫ 2π
B
∫
= 2πvol(B)
0
∫
B
B∩L
∫
1
cos ɛ (r)
∫B
dθdΣdx = 2πC ∫
G C n,n−1
CH n−1 (ɛ)
L C (n−1)[x]
L C 1
1
cos ɛ (r) dΣdx
cos 2 ɛ(r)
cos ɛ (r) dp n−1dG n,n−1 dx
cos ɛ (r)dx = +∞.
5.2 Variation of the measure of planes meeting a regular domain
At Chapter 4 we give an expression for the measure of complex r-planes meeting a regular
domain in CK n (ɛ). Now, we give a generalization of this result for the space of (k, p)-planes
in C n , and for the space of totally real k-planes in CK n (ɛ).
First, we need the expression of the density of the space of (k, p)-planes with respect to
the forms ω ij defined at (1.13).

5.2 Variation of the measure of planes meeting a regular domain 85
Lemma 5.2.1.
1. In C n , the space L k,p is a homogeneous space and
L k,p
∼ = U(n) ⋉ C n /(U(p) × O(k − 2p) × U(n − k + p) ⋉ R n ).
Let {g; g 1 , g 2 , ..., g 2n−1 , g 2n } with g 2i = Jg 2i+1 be a J-moving frame adapted to a (k, p)-
plane in g such that {g 1 , Jg 1 , ..., g p , Jg p , g 2p+1 , ..., g 2k−1 } expand the tangent space to the
(k, p)-plane. The invariant density of L k,p is given by
∣ ∣∣∣∣∣ ∧ ∧
dL k,p =
ω i ω ji (5.2)
∣
i
where i ∈ {2p + 2, 2p + 4, ..., 2k, 2k + 1, 2k + 2, ..., 2n} and j ∈ {1, 3, ..., 2p − 1, 2p + 1, 2p +
3, ..., 2k − 1}.
2. In CK n (ɛ), ɛ ≠ 0, the space of complex p-planes L C p and the space of totally real k-planes
L R k
are homogeneous spaces and
j,i
L C p ∼ = U ɛ (n)/(U ɛ (p) × U(n − p)),
where
L R k ∼ = U ɛ (n)/(O ɛ (k) × U(n − k)),
U ɛ (n) =
{ U(1 + n), if ɛ > 0,
U(1, n), if ɛ < 0.
, O ɛ (k) =
{ O(1 + k), if ɛ > 0,
O(1, k), if ɛ < 0.
Moreover, fixed a J-moving frame as in the previous statement, the expression (5.2)
remains true.
Proof. 1. By Lemma 1.5.1 we have that the isometry group of C n acts transitively over
J-basis. Thus, there exists an isometry that carries a fixed (k, p)-plane to another.
The isotropy group of a (k, p)-plane in C n is isomorphic to U(p)×O(k−2p)×U(n−k+p)
since (k, p)-planes in C n are totally geodesic submanifolds and the tangent space at each
point is isometric to C p ⊕ R k−2p .
The density can be obtained using the theory of moving frames that we have discussed
in Section 1.3.
2. The arguments for the previous case are also valid, since we restrict to totally geodesic
submanifolds.
The following proposition is a generalization of the Proposition 4.2.1 for any (k, p)-plane
in C n .
Proposition 5.2.2. Let Ω ⊂ C n be a regular domain, X a smooth vector field defined at C n
with φ t the flow associated to X and Ω t = φ t (Ω). Then, in the space of (k, p)-planes L k,p in
C n , it is satisfied
∫
∫
d
dt∣ χ(Ω t ∩ L k,p )dL k,p = 〈∂φ/∂t, N〉
σ k (II| V )dV dx
t=0
∫L k,p
∂Ω
G n,k,p (T x∂Ω)
where N is the outward normal field at ∂Ω and σ k (II| V ) denotes the k-th symmetric elementary
function of II restricted to V ∈ G n,k,p (T x ∂Ω), the Grassmanian of the (k, p)-planes contained
in the tangent space of ∂Ω at x.

86 Other Crofton formulas
It is also satisfied the following extension of Proposition 4.2.1 for totally real planes in
CK n (ɛ).
Proposition 5.2.3. Let Ω ⊂ CK n (ɛ) be a regular domain, X a smooth vector field defined at
CK n (ɛ) with φ t the flow associated to X and Ω t = φ t (Ω). Then, in the space of totally real
k-plane L R k in CKn (ɛ), it is satisfied
∫
d
dt∣ t=0
L R k
∫
∫
χ(Ω t ∩ L k,p )dL k,p = 〈∂φ/∂t, N〉
σ k (II| V )dV dx
∂Ω
G n,k,0 (T x∂Ω)
where N is the outward normal field at ∂Ω and σ k (II| V ) denotes the k-th symmetric elementary
function of II restricted to V ∈ G n,k,0 (T x ∂Ω), the Grassmanian of the totally real k-planes
contained in the tangent space of ∂Ω at x.
Proof. This proof is analog to the proof of Proposition 4.2.1, since the expression for the density
of the space of totally real planes in (5.2) holds. We just have to modify the construction of
the map γ in (4.7).
For every x ∈ ∂Ω consider the curve c(t) = ϕ t (x). For every t, let D c(t) = 〈N c(t) , JN c(t) 〉 ⊥ ⊂
dϕ t (T x ∂Ω) the complex hyperplane tangent to ϕ t (∂Ω) at c(t). If ∇ ∂t denotes the covariant
derivative of CK n (ɛ) along c(t), we define
∇ D ∂t X(t) = π t(∇ ∂t X(t))
where π t : T c(t) CK n (ɛ) → D c(t) denotes the orthogonal projection. Given a vector X ∈ T x ∂Ω,
there exists a unique vector field X(t) defined along c(t) such that ∇ D ∂tX(t) = 0 (it can be
proved in the same way as the existence of the usual parallel translation). This define a linear
map ψ t : D x → D c(t) , which preserves the complex structure J since
∇ D ∂t JX(t) = π t(∇ ∂t JX(t)) = π t (J∇ ∂t X(t)) = Jπ t (∇ ∂t X(t)).
Finally, we extend ψ t linearly to ψ t : T x ∂Ω → dϕ t (T x ∂Ω) such that ψ t (JN x ) = JN c(t) . This
map takes totally real planes into totally real planes. So, we can define the new map γ as
γ : G n,k,p (T ∂Ω) × (−ɛ, ɛ) −→ L k,p
((x, V ), t) ↦→ exp φt(x) ψ t (V ) .
5.3 Measure of real geodesics in CK n (ɛ)
The following result, obtained straightforward from the last proposition, states that in complex
space forms, the measure of real geodesics meeting a regular domain is a multiple of the area
of the domain (as in real space forms).
Theorem 5.3.1. Let Ω ⊂ CK n (ɛ) be a regular domain, let X be a smooth vector field over
CK n (ɛ), let φ t be the flow associated to X and let Ω t = φ t (Ω). Then
∫
d
dt∣ χ(Ω t ∩ L 1 )dL R 1 = O 2n+1 ( ˜B 2n−2,n−2 (Ω) + ˜Γ 2n−2,n−1 (Ω))
t=0
L R 1
and
∫
L R 1
χ(Ω ∩ L 1 )dL R 1 = ω 2n µ 2n−1,n−1 (Ω) = ω 2n
2 vol(∂Ω).

5.4 Measure of real hyperplanes in C n 87
Proof. From Proposition 5.2.2 we have
∫
∫
d
dt∣ χ(Ω t ∩ L 1 )dL R 1 =
t=0
L R 1
∂Ω
∫
〈∂φ/∂t, N〉
σ 1 (II| V )dV dx.
G R n,1,0 (Tx∂Ω)
Let us study the integral with respect to the Grassmanian of geodesics in the tangent space
of each point x ∈ ∂Ω. We denote by {f 1 , . . . , f 2n−1 } the principal directions at x. Then,
∫
σ 1 (II| V )dV = 1 ∫
k n (v)dv = 1 2n−1
∑
∫
〈v, f i 〉 2 k i dv
G R n,1,0 (Tx∂Ω) 2 S 2n−1 2
i=1
S 2n−1
= 1 2n−1
∑
∫
k i 〈v, f i 〉 2 dv (3.1.2)
= O 2n−1
∑
2n−1
k i = O 2n−1(2n − 1)
tr(II).
2
S 2n−1 4n
4n
i=1
Thus, by Examples 2.4.20.3 and 2.4.20.4
∫
d
dt∣ χ(Ω t ∩ L 1 )dL R 1 = O ∫
2n−1(2n + 1)
〈X, N〉tr(II)dx
t=0 L R 4n
1
∂Ω
= O ∫
(
2n−1
〈X, N〉
4n
= O 2n−1
4 n!
=
ω 2n
2 (n − 1)!
N(Ω)
i=1
1
(n − 1)! γ ∧ θn−1 2 +
( ∫
〈X, N〉γ ∧ θ2 n−1 + (n − 1)
N(Ω)
N(Ω)
(˜Γ′ 2n−2,n−1 (Ω) + (n − 1) ˜B 2n−2,n−2(Ω))
′
= O 2n+1 ( ˜B 2n−2,n−2 (Ω) + ˜Γ 2n−2,n−1 (Ω)).
)
1
(n − 2)! β ∧ θ 1 ∧ θ2
n−2
∫
)
〈X, N〉β ∧ θ 1 ∧ θ n−2
In C n , the valuation ∫ L
χ(∂Ω∩L R 1 )dL 1 has degree 2n−1, so, it is a multiple of B 2n−1,n−1 (Ω),
1
which has variation (cf. Proposition 4.1.7)
δ X µ 2n−1,n−1 (Ω) = c n,2n−1,n−1 (2c −1
n,2n−2,n−1˜Γ 2n−2,n−1 (Ω) + c −1
n,2n−2,n−2 (n − 1) ˜B 2n−2,n−2 (Ω))
= c n,2n−2,n−1 (˜Γ ′ 2n−2,n−1(Ω) + (n − 1) ˜B 2n−2,n−2(Ω))
′
1
= (˜Γ ′
(n − 1)!ω
2n−2,n−1(Ω) + (n − 1) ˜B 2n−2,n−2(Ω)).
′
1
Therefore, comparing both variations we get the stated result in C n . The same expression
holds for ɛ ≠ 0 since the variation of µ 2n−1,n−1 does not depend on ɛ.
The relation with the volume of the boundary of the domain is obtained from the relation
of β 2n−1,n−1 with the second fundamental form given in Example 2.4.20.5.
5.4 Measure of real hyperplanes in C n
The measure of real hyperplanes intersecting a regular domain in C n also follows immediately
from Proposition 5.2.2. This particular case has interest by its own since real hyperplanes are
submanifolds of codimension 1.
Theorem 5.4.1. Let Ω ⊂ C n be a regular domain, X a smooth vector field over C n , φ t the
flow associated to X and Ω t = φ t (Ω). Then
d
dt∣ χ(Ω t ∩ L 2n−1,n−1 )dL 2n−1,n−1 = O 2n+1˜Γ0,0 (Ω)
t=0
∫L 2n−1,n−1
and
∫
L 2n−1,n−1
χ(Ω ∩ L 2n−1,n−1 )dL 2n−1,n−1 = ω 2n−1 µ 1,0 (Ω).
2

88 Other Crofton formulas
Proof. From Proposition 4.2.1 we have
∫ ∫
d
dt∣ χ(Ω t ∩ L 2n−1,n−1 )dL 2n−1,n−1 = 〈X, N〉
σ 2n−1 (II| V )dV dx
t=0
∫L 2n−1,n−1 ∂Ω 0 G n,2n−1,n−1 (T
∫
x∂Ω)
= 〈X, N〉σ 2n−1 (II)dx
∫∂Ω
1
= 〈X, N〉
(n − 1)! γ ∧ θn−1 0
N(Ω)
= 2c−1 n,0,0,
(n − 1)! ˜Γ 0,0 (Ω).
In C n , the valuation ∫ L 2n−1,n−1
χ(∂Ω ∩ L 2n−1,n−1 )dL 2n−1,n−1 has degree 1, thus, it is a
multiple of µ 1,0 , which has variation
δ X µ 1,0 = 2c n,1,0 c −1
n,0,0˜Γ 0,0 .
Then, comparing both expressions we obtain the result.
Remark 5.4.2. From Example 2.4.20, it follows the equality
µ 1,0 (Ω) = 1 det(II| D ) =
ω 2n−1
∫∂Ω
1 M
ω
2n−2(∂Ω).
D
2n−1
On the other hand, there is just one linearly independent valuation in the space of continuous
translation and U(n)-invariant valuations in C n of degree 1. Thus,
µ 1,0 (Ω) = cM 2n−2 (∂Ω),
and the measure of real hyperplanes in C n meeting a regular domain is a multiple of the
so-called “mean width”, as in the Euclidean space.
5.5 Measure of coisotropic planes in C n
A subspaces of C n is called coisotropic if its orthogonal is a totally real plane.
Lemma 5.5.1. The (2n − p, n − p)-planes in C n are the coisotropic planes.
Proof. If L ∈ L 2n−p,n−p , then L ⊥ has dimensió 2n − (2n − p) = p. The dimension of the
maximal complex subspace contained in L ⊥ is n − (n − p) − p = 0. Thus, L ⊥ is a totally real
plane.
Reciprocally, if L ⊥ is a totally real p-plane, then L has dimension 2n − p and the maximal
complex subspace has dimension n − p.
Lemma 5.5.2. Let S ⊂ C n be a hypersurface and L ∈ L 2n−p,n−p , p ∈ {1, . . . , n}, be a
(2n−p, n−p)-plane tangent in S at p. If N denotes a normal vector to S at x, then JN ∈ T x L.
Proof. As L is a (2n−p, n−p)-plane, we can consider, at each point, a basis of its tangent space
of the form {e 1 , Je 1 , . . . , e n−p , Je n−p , e n−p+1 , e n−p+2 , . . . , e n }, in a way such that Je i ⊥T x L for
i ∈ {n−p+1, . . . , n}. Moreover, we can complete this basis to a basis of T x C n with the vectors
{J en−p+1 , . . . , Je n }.
On the other hand, at x ∈ L ∩ S, is it satisfied T x L ⊂ T x S, thus, N⊥T x L, i.e.
〈N, e i 〉 = 0, ∀ i ∈ {1, . . . , n}, (5.3)
〈N, Je j 〉 = 0,
∀ j ∈ {1, . . . , n − p}.
Now, if JN = ∑ n
i=1 α ie i + ∑ n
i=1 β iJe i , then N = − ∑ n
i=1 α iJe i + ∑ n
i=1 β ie i . Using (5.3)
we get JN = ∑ n
i=n−p+1 α ie i . Thus, JN ∈ T x L.

5.5 Measure of coisotropic planes in C n 89
From this lemma, we can prove the following result.
Theorem 5.5.3. Let Ω ⊂ C n be a regular domain, X a smooth vector field defined in C n , φ t
the flow associated to X and Ω t = φ t (Ω). Then,
∫
d
dt∣ χ(Ω t ∩ L)dL = vol(G ( )
n,2n−p,n−p)ω 2n−p+1
n
(2n − p + 2)
−1· (5.4)
t=0 L 2n−p,n−p
(n − 1)!
p − 1
and
∫
·
⌊ p−1
2 ⌋ ∑
q=max{0,p−n−1}
4 q−p+1 ( ) 2n + 2q − 2p + 1 −1˜Γp−1,q (Ω),
(2n + 2q − 2p + 3) n + q − p + 1
χ(Ω ∩ L)dL = vol(G n,2n−p,n−p)ω 2n−p
L 2n−p,n−p
(n − 1)!
·
⌊ p 2 ⌋ ∑
q=max{0,p−n}
( ) n −1·
p − 1
( ) 2n + 2q − 2p − 1 −1
4 q−p
n + q − p − 1 2n + 2q − 2p + 1 µ p,q(Ω).
Proof. First of all, we prove that for the space of coisotropics planes, the variation of the
measure does not have contribution in ˜B p,q . From Proposition 5.2.2 we have
∫
∫
d
dt∣ χ(Ω t ∩ L)dL = 〈∂φ/∂t, N〉
σ 2n−p (II| V )dV dx
t=0
∫L 2n−p,n−p
∂Ω
G n,2n−p,n−p (T x∂Ω)
but each V ∈ G n,2n−p,n−p (T x ∂Ω), by the previous lemma, contains the JN direction (with N
the outward normal vector to ∂Ω at x), so that II| V always contains the entry corresponding
to the normal curvature of the direction JN. From Lemma 2.4.18 we have that only the
polynomials obtained from φ ∗ (γ k,q ) contain this entry of the second fundamental form.
∫
In order to find the constants, we solve a linear system. First, note that the functional
L 2n−p,n−p
χ(Ω ∩ L)dL is a valuation in C n with homogeneous degree p. Thus, it can be
expressed as a linear combination of the Hermitian intrinsic volumes with the same degree
∫
L 2n−p,n−p
χ(Ω ∩ L)dL =
⌊ p−1
2 ⌋ ∑
q=max{0,p−n}
A p,q µ p,q (Ω) (5.5)
for some A p,q , which we want to determine.
Taking the variation in both sides, we find the value of these constants. By Proposition
4.1.7, the variation on the right hand side of (5.5) is
⌊ p 2 ⌋−1 ∑
q=max{0,p−n−1}
(A p,q 2c n,p,q c −1
n,p−1,q (p − 2q)2 (5.6)
− A p,q+1 2c n,p,q+1 c −1
n,p−1,q (n − p + q + 1)(q + 1))˜Γ p−1,q
+ (A p,q+1 2c n,p,q+1 c −1
n,p−1,q (n − p + q + 3/2)(q + 1)
− A p,q 2c n,p,q c −1
n,p−1,q (p − 2q)(p − 2q − 1)) ˜B p−1,q
+ A p,⌊
p
2 ⌋ 2c n,p,⌊
p
2 ⌋ c n,p−1,⌊
p
2 ⌋ (p − 2⌊ p 2 ⌋)2˜Γp−1,⌊ p
2 ⌋ .
Imposing that the variation vanishes on ˜B p−1,q we get some equations, from which we obtain
the relations
A p,q+1 = n − p + q + 1
(n − p + q + 3/2) A p,q. (5.7)

90 Other Crofton formulas
So, each A p,q+1 is a multiple of A p,max{0,p−n} . To find this value, we need another equation,
obtained taking II| D = λId, λ ∈ R + , and equation the expression (5.6) with the variation of
the Proposition 5.2.2. Then, for each (n, r) we have a compatible linear system, since constant
in (5.5) exist. Moreover, by (5.7) they are unique. Doing so, we get, in the same way as in the
proof of Proposition 4.3.1, the desired result. Finally, we get the variation substituting the
obtained values of A p,q at (5.6).
An interesting particular case of the last theorem is the case of Lagrangian planes. This is
the case stated by Alesker [Ale03] as a remarkable case of study. From the previous theorem
we can give explicitly the constant in the theorem of Alesker reproduced at 2.3.5, but with
respect to the Hermitian intrinsic volumes defined by Bernig-Fu, and not directly by the bases
defined by Alesker.
Corollary 5.5.4. Let Ω be a regular domain in C n with piecewise smooth boundary. Then,
∫
L R n
χ(Ω ∩ L)dL = vol(G n,n,0)ω n
n!
⌊ n−1
2 ⌋
∑
q=0
where L R n denotes the space of Lagrangian planes in C n .
( ) 2q − 1 −1
4 q−n
q − 1 2q + 1 µ n,q(Ω).
5.6 Measure of Lagrangian planes in CK n (ɛ)
Using the same techniques as in Chapter 4, it can be proved the following result.
Theorem 5.6.1. Let Ω ⊂ CK n (ɛ) be a regular domain, X a smooth vector field defined at
CK n (ɛ), φ t the flow associated to X and Ω t = φ t (Ω). Then,
∫
d
dt∣ t=0
L R n
and
if n is odd
χ(Ω t ∩ L)dL = vol(G n,n,0 )ω n+1
(n + 2)
n!
⌊ n−1
2 ⌋ ∑
q=0
4 q−n+1
2q + 3
( ) 2q + 1 −1˜Γn−1,q (Ω),
q + 1
∫
L R n
χ(Ω ∩ L)dL = vol(G n,n,0)ω n
n!
n−1
2∑
q=0
( ) 2q − 1 −1
4 q−n
q − 1 2q + 1 µ n,q(Ω), (5.8)
and if n is even
∫
χ(Ω ∩ L)dL = vol(G n,n,0)
· (5.9)
L R n!
n
⎛
n
2∑
( )
· ⎝ 2q − 1 −1
4 q−n n
⎞
ω n
2∑
( ) n −1
q − 1 2q + 1 µ n,q(Ω) + ɛ i 2 −n+1 ω n−2i
n
2 + i µ
n + 1 n+2i,
n
2 +i (Ω) ⎠ .
q=0
i=1
Proof. In the same way as in the proof of Theorem 4.3.5, it is enough to prove that the
variation in both sides coincides.
The variation on the left hand side of (5.9) and (5.8) coincides and is independent on ɛ.
Thus, it coincides with the variation in (5.4).
We compute the variation on the right hand side by using Proposition 4.1.7. Here, we just
reproduce the computations when n is odd. For n even, a similar, but longer study can be
done to verify expression (5.9).

5.6 Measure of Lagrangian planes in CK n (ɛ) 91
Denote by E n (Ω) the right hand side of (5.8). Then, by (5.4) we have
δ X E n (Ω)= vol(G n,n,0)ω n
n!
n−1
2∑
q=0
( ) 2q − 1 −1
4 q−n
q − 1 2q + 1 2c n,n,q·
(
· c −1
n,n−1,q (n − 2q)2˜Γn−1,q − c −1
n,n−1,q−1 q2˜Γn−1,q−1
+ c −1
n,n−1,q−1 q(q + 1/2) ˜B n−1,q−1 − c n,n−1,q (n − 2q)(n − 2q − 1) ˜B n−1,q
+ ɛ(c −1
n,n+1,q+1 (n − 2q)(n − 2q − 1) ˜B n+1,q+1 − c −1
n,n+1,q q(q + 1/2) ˜B n+1,q )
n−3 (
)
= 2 vol(G n,n,0)ω n
2∑ c n,n,q 4 q−n (n − 2q) 2
{ {
n!
(2q + 1) ( )
2q−1
− c n,n,q+14 q−n+1 (q + 1) 2
q=0
q−1
(2q + 3) ( )
2q+1
q
(
)
c n,n,q+1 4 q−n+1 (q + 1)(q + 3/2)
+
(2q + 3) ( )
2q+1
− c n,n,q4 q−n (n − 2q)(n − 2q − 1)
q
(2q + 1) ( )
2q−1
q−1
+ɛ
(
2∑ c n,n,q−1 4 q−n−1 (n − 2q + 2)(n − 2q + 1)
(2q − 1) ( )
2q−3
− c n,n,q4 q−n q(q + 1 2 )
)
q−2
(2q + 1) ( )
2q−1
q−1
n−1
q=1
)
c −1
n,n−1,q ˜Γ n−1,q
c −1
n,n−1,q ˜B n−1,q }
c −1
n,n+1,q ˜B n+1,q }.
In order to prove the result, it suffices to prove that this expression is independent on ɛ. As
for ɛ = 0 we know that δ X E n (Ω) coincides with (5.4), we get the result.
Now, to prove the independence of ɛ, we collect the coefficient for each ˜B n−1,q and ˜B n+1,q ,
and we prove that they vanish.

Appendix
This appendix contains a constructive proof of Theorems 4.3.5 and 4.4.1.
Proof of Theorem 4.3.5
We prove that it is possible to find constants α k,q such that
∫
L C r
χ(Ω ∩ L r )dL r = ∑ k,q
⌊n/2⌋
∑
α k,q B k,q (Ω) + α 2j,j Γ 2j,j (Ω) + α 2n,n vol(Ω)
j=1
(A.10)
where max{0, k − n} ≤ q < k/2 ≤ n.
For ɛ = 0, the existence of these constants follows from the fact that Hermitian intrinsic
volumes constitute a basis of smooth valuations. If ɛ ≠ 0, we cannot ensure this fact. Anyway,
we find the value of the previous constants imposing that the variation in both sides of (A.10)
coincides. This is enough to prove (A.10). Indeed, take a deformation Ω t of Ω such that Ω t
converges to a point. Then, both sides of (A.10) have the same variation and it vanishes in
the limit.
The variation of the left hand side of (A.10) is given in Corollary 4.3.3. The variation of
the right hand side can be computed using Proposition 4.1.7 and δ X vol = 2 ˜B 2n−1,n−1 .
In the variation of the left hand side, just appear the terms { ˜B 2n−2r−1,q } q . Thus, the
variation of the right hand side can only have these terms. On the other hand, the variation
of a Hermitian intrinsic volume B k,q with k even (resp. odd) has only terms ˜B a,b and ˜Γ a ′ ,b ′
with a, a ′ odd (resp. even) (cf. Proposition 4.1.7). As the variation of the left hand side has
only non-vanishing terms with odd subscript, we just consider the valuations with first even
subscript. Doing also the change in (4.12), expression (A.10) reduces to
∫
L C r
n−1
∑
(
k=1
n−1
∑
χ(Ω ∩ L r )dL r = (
k=1
∑k−1
q=max{0,2k−n}
C 2k,q B 2k,q ′ (Ω) + D 2k,kΓ ′ 2k,k (Ω)) + dvol(Ω). (A.11)
Now, we start the study to find constants C k,q , D 2q,q , d such that
∑k−1
q=max{0,2k−n}
⎛
= vol(GC n−1,r )ω 2r+1(r + 1)
)( ⎝
n
r)
( n−1
r
C 2k,q δB 2k,q ′ (Ω) + D 2k,kδΓ ′ 2k,k (Ω) + dδvol(Ω)
n−r−1
∑
q=max{0,n−2r−1}
⎞
( )
2n − 2r − 2q − 1 1
n − r − q 4 ˜B ′ n−r−q−1 2n−2r−1,q(Ω) ⎠ .
By Proposition 4.1.7, this equation gives rise to a linear system. We write this linear system
in matrix form Ax = b. Consider the vector of unknowns as
x t = (C 2,0 , D 2,1 , C 4,0 , C 4,1 , D 4,2 , . . . , C 2c,max{0,2c−n} , . . . , D 2c,c , . . . , C 2n−2,n−2 , D 2n−2,n−1 , d).
93

94 Appendix
Vector b contains the coefficient of ˜B ′ k,q , ˜Γ ′ k,q
given in (4.23), that is
b t = vol(GC n−1,r ) ( ( )( ) ( )( n − r r + 1 n − r r + 1
n! ( )
n−1
0, . . . , 0,
,
1 1 2 2
r
) 1
2 , . . . , ( n − r
r + 1
)( r + 1
r + 1
) )
r + 1
4 r , 0, . . . , 0 .
Note that b has all entries null except the ones corresponding to ˜B 2n−2r−1,q ′ .
The coefficients of the matrix A contain the variation of each B
2k,q ′ and Γ′ 2q,q with respect
to ˜B r,s ′ and ˜Γ ′ r,s. We denote by (δB
k,q ′ , ˜B r,s ), the coefficient of ˜Br,s in the variation of the
valuation B
k,q ′ . By Proposition 4.1.7
(δB
k,q ′ , ˜B r,s) ′ =
(δB
k,q ′ , ˜Γ ′ r,s) =
(δΓ ′ 2q,q , ˜B r,s) ′ =
(δΓ ′ 2q,q , ˜Γ ′ r,s) =
⎧
⎪⎨
⎪⎩
⎧
⎨
⎩
⎧
⎪⎨
⎪⎩
⎧
⎨
⎩
2q(n + q − k + 1/2), if r = k − 1, s = q − 1
−2(k − 2q)(k − 2q − 1), if r = k − 1, s = q
2ɛ(k − 2q)(k − 2q − 1), if r = k + 1, s = q + 1
−2ɛ(n − k + q)(q + 1/2), if r = k + 1, s = q
0, otherwise.
(k − 2q) 2 , if r = k − 1, s = q
−(n + q − k)q, if r = k − 1, s = q − 1
0, otherwise.
4q(n − q + 1/2), if r = 2q − 1, s = q − 1
−4ɛ((n − q)(2q + 3/2) − (q + 1)/2), if r = 2q + 1, s = q
4ɛ 2 (n − q − 1)(q + 3/2), if r = 2q + 3, s = q + 1
0, otherwise.
−2(n − q)q, if r = 2q − 1, s = q − 1
2ɛ(n − q − 1)(q + 1), if r = 2q + 1, s = q
0, otherwise.
Each column of the matrix A contains the variation of a valuation B
2k,q ′ , Γ′ 2q,q or the volume.
We take the valuations B
2k,q ′ , Γ′ 2q,q in the same order as in the vector b. (The volume corresponds
to the last column.) That is, the columns of A contain the variation of the valuations
in the following order
(δB ′ 2,0, δΓ ′ 2,1, δB ′ 4,0, δB ′ 4,1, δΓ ′ 4,2, . . . , δB ′ 2n−2,n−2, δΓ ′ 2n−2,n−1, δvol).
We denote
δB ′ 2k,· = (δB′ 2k,max{0,2k−n} , δB′ 2k,max{0,2k−n}+1 , . . . , δB′ 2k,k−1 ),
δΓ ′ 2k,· = δΓ′ 2k,k ,
˜B ′ 2k+1,· = ( ˜B ′ 2k+1,max 0,2k−n+1 , ˜B ′ 2k+1,max 0,2k−n+1+1 , . . . , ˜B ′ 2k+1,k )t ,
˜Γ ′ 2k+1,· = (˜Γ ′ 2k+1,max 0,2k−n+1 , ˜Γ ′ 2k+1,max 0,2k−n+1+1 , . . . , ˜Γ ′ 2k+1,k )t .

Proof of Theorem 4.3.5 95
Then, A has the following boxes structure
δB 2,· ′ δΓ ′ 2,· δB 4,· ′ δΓ ′ 4,· δB 6,· ′ δΓ ′ 6,· δB 8,· ′ δΓ ′ 8,· · · · δB 2n−4,· ′ δΓ ′ 2n−4,· δB 2n−2,· ′ δΓ ′ 2n−2,· δvol
˜B 1,· ′ ∗ ∗
˜Γ ′ 1,· ∗ ∗
˜B 3,· ′ ∗ ɛ ∗ ɛ ∗ ∗
˜Γ ′ 3,· ∗ ɛ ∗ ∗
˜B 5,· ′ ∗ ɛ ∗ ɛ ∗ ɛ ∗ ∗
˜Γ ′ 5,· ∗ ɛ ∗ ∗
˜B 7,· ′ ∗ ɛ ∗ ɛ ∗ ɛ ∗ ∗
˜Γ ′ 7,· ∗ ɛ ∗ ∗
˜B 9,· ′ ∗ ɛ ∗ ɛ ∗ ɛ
˜Γ ′ 9,·
∗ ɛ
.
˜B 2n−3,· ′ ∗ ɛ ∗ ɛ ∗ ∗
˜Γ ′ 2n−3,· ∗ ɛ ∗ ∗
˜B 2n−1,· ′ ∗ ɛ ∗ ɛ ∗ ɛ ∗
We denoted by ∗ the boxes of A with non-null coefficients and independent of ɛ, and by ∗ ɛ the
boxes of A with non-null coefficients (for ɛ ≠ 0) and multiples of ɛ.
The structure by boxes of the linear system given by A suggests the method of resolution:
we start with the top box, and we get the value of variables C 2,q and D 2,1 . Then we solve the
next bloc with rows ˜B ′ 3,·, ˜Γ ′ 3,·, using the value of variables C 2,q and D 2,1 . We can continue this
process, so that, once we know the value of variables C 2k,q and D 2k,k , we substitute it on the
equations given by the rows ˜B ′ 2k+1,·, ˜Γ ′ 2k+1,·.
Recall that the independent vector b has all terms null except the ones corresponding to
˜B ′ 2n−2r−1,q . Thus, the linear system is homogeneous for the first equations until ˜B′ 2n−2r−2,q ,
and we can take C k,q = D 2q,q = 0 whenever k ≤ 2n − 2r − 1.
By Theorem 4.3.1 we have a solution for the system Ax = b when ɛ = 0. This solution
has C 2n−2r,q and D 2n−2r,n−r as non-null terms, and satisfies the equations until rows
˜B 2n−2r,·, ′ ˜Γ ′ 2n−2r,· also for ɛ ≠ 0.
So, we consider for all ɛ ∈ R and for all a ∈ {1, ..., min{n − r, r}}
C 2n−2r,n−r−a = vol(GC n−1,r )
4 a n!
D 2n−2r,n−r = vol(GC n−1,r )
2 n!
( ) n − 1 −1 ( n − r
r a
( ) n − 1 −1
.
r
)( r
a)
,
Now, we go on with the resolution of the linear system in CK n (ɛ). We study for each
c ∈ {n − r + 1, . . . , n} the submatrix of A with all rows ˜B 2c−1,q ′ , ˜Γ ′ 2c−1,q . This matrix has the
following non-zero columns of A: δΓ ′ 2c−4,c−2 , δB′ 2c−2,·, δΓ′ 2c−2,c−1 , δB′ 2c,· and δΓ′ 2c,c .
Suppose that we know the value of D 2c−4,c−2 , C 2c−2,q and D 2c−2,c−1 . Then, we can substitute
them in the equations given by the rows corresponding to ˜B 2c−1,q , ˜Γ 2c−1,q . We get
equations with C 2c,q and D 2c,c as unknowns. If we denote i = max{0, 2c − n}, the matrix of
the coefficients for the obtained equations (which corresponds to a matrix bloc of A indepen-

96 Appendix
dent of ɛ) is
δB 2c,i δB 2c,i+1 δB 2c,i+2 · · · δB 2c,c−2 δB 2c,c−1 δΓ 2c,c
˜B 2c−1,i −4c(2c − 1) 2(n − 2c + 3/2)
˜B 2c−1,i+1 −2(2c − 2)(2c − 3) 4(n − 2c + 5/2)
˜B 2c−1,i+2 −2(2c − 4)(2c − 5)
.
˜B 2c−1,c−2 −24 2(c − 1)(n − c − 1 ) 2
˜B 2c−1,c−1 −4
˜Γ 2c−1,0 (2c) 2 −(n − 2c + 1)
4c(n − c + 1 ) 2
˜Γ 2c−1,1 (2c − 2) 2 −2(n − 2c + 2)
˜Γ 2c−1,2 (2c − 4) 2
.
˜Γ 2c−1,c−2 4 2 −(c − 1)(n − 1)
˜Γ 2c−1,c−1 4 −2c(n − c)
The independent term is obtained from the initial independent term b (which in these
cases is always zero) and from the part of the initial equation in which we substituted the
value of D 2c−4,c−2 , C 2c−2,q , D 2c−2,c−1 . Comparing the box structure of A on page 95 and its
coefficients on page 94, we obtain that the independent term of this new linear system has
zero the terms ˜Γ ′ 2c−1,q with q ∈ {max{0, 2c − n}, ..., c − 2}, and the term ˜Γ 2c−1,c−1 equals to
2ɛ(n − c)cD 2c−2,c−1 .
Now, we consider the equations given by rows ˜Γ ′ 2c−1,q and ˜B 2c−1,max{0,2c−n} in the previous
matrix, which give a compatible linear system with one solution. The independent terms of
the equation given by equation ˜B 2c−1,max{0,2c−n} is ɛ(n−2c+2)C 2c−2,max{0,2c−n−2} . The other
ones are zero. Solving this system we get the variables C 2c,max{0,2c−n} and D 2c,c in terms of
C 2c−2,max{0,2c−n−2} and D 2c−2,c−1 , which we suppose known.
In order to avoid considering the maximum max{0, 2c − n − 2} we distinguish two cases.
First stage: 2c ≤ n. This case appears if 2r > n (since c ∈ {n − r + 1, . . . , n}).
The linear system we have to solve is given by the augmented matrix
⎛
⎞
−4c(2c − 1) 2(n − 2c + 3/2) ɛ(n − 2c + 2)C 2c−2,0
(2c) 2 −(n − 2c + 1)
(2c − 2) 2 −2(n − 2c + 2)
.
⎜
.. ⎟
⎝
⎠
−(c − 1)(n − c − 1)
4 −2c(n − c) −2ɛc(c − n)D 2c−2,c−1
with variables {C 2c,0 , C 2c,1 , . . . , C 2c,c−1 , D 2c,c }.
From the first two equations we obtain
ɛ(n − 2c + 2)(n − 2c + 1)
C 2c,0 = C 2c−2,0 ,
4c(n − c + 1)
ɛ(n − 2c + 2)c
C 2c,1 = C 2c−2,0 .
(n − c + 1)
For every q ∈ {0, ..., c − 2} the following relations are satisfied
and we get
C 2c,q+1 =
(2c − 2q) 2 C 2c,q = (q + 1)(n − 2c + q + 1)C 2c,q+1 ,
4C 2c,c−1 − 2c(n − c)D 2c,c = −2ɛc(n − c)D 2c−2,c−1 ,
ɛ4 q (n − 2c + 2)!c!(c − 1)!
(n − c + 1)(q + 1)!(n − 2c + q + 1)!(c − q − 1)!(c − q − 1)! C 2c−2,0,
D 2c,c = ɛ(D 2c−2,c−1 +
2(n − 2c + 2)!(c − 1)!4c−2
C 2c−2,0 ).
(n − c + 1)!
(A.12)

Proof of Theorem 4.3.5 97
As the known constants are C 2n−2r,q and D 2n−2r,n−r we write the previous ones in terms
of these, using the recurrence we just obtained.
C 2c,0 =
ɛ(n − 2c + 2)(n − 2c + 1)
C 2c−2,0
4c(n − c + 1)
= ɛc−(n−r) (n − 2c + 2)(n − 2c + 1) · ... · (n − 2n + 2r − 2 + 2)(n − 2n + 2r − 2 + 1)
4 c−(n−r) c(c − 1) · ... · (n − r + 1)(n − c + 1)(n − c + 2) · ... · (n − (n − r))
(A.13)
= ɛc−(n−r) (2r − n)!(n − r)!(n − c)!vol(G C n−1,r ) ( ) n − 1 −1 ( ) r
4 c−(n−r) (n − 2c)!c!r!4 n−r n!
r n − r
= ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1 ( ) n − c
4 c ,
n!
r c
C 2c,q+1 =
C 2n−2r,0
ɛ(n − 2c + 2)4 q (n − 2c + 1)!c!(c − 1)!
(n − c + 1)(q + 1)!(n − 2c + q + 1)!(c − q − 1)!(c − q − 1)! C 2c−2,0 (A.14)
= ɛ(n − 2c + 2)4q (n − 2c + 1)!c!(c − 1)!ɛ c−1−(n−r) vol(G C n−1,r )(n − c + 1)!
(n − c + 1)(q + 1)!(n − 2c + q + 1)!((c − q − 1)!) 2 4 c−1 n!(c − 1)!(n − 2c + 2)!
ɛ c−(n−r) c!(n − c)!vol(G C n−1,r
=
)
( ) n − 1 −1
4 c−q−1 (q + 1)!(n − 2c + q + 1)!(c − q − 1)!(c − q − 1)!n! r
= ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1 ( )( )
c n − c
4 c−q−1 ,
n! r q + 1 c − q − 1
( n − 1
)
2(n − 2c + 2)!(c − 1)!4c−2
D 2c,c = ɛ
(D 2c−2,c−1 + C 2c−2,0
(n − c + 1)!
(
2(n − 2c + 2)!(c − 1)!4c−2 ɛ c−1−(n−r) vol(G C ( ) )
n−1,r )(n − c + 1)! n − 1 −1
= ɛ D 2c−2,c−1 +
(n − c + 1)! 4 c−1 n!(c − 1)!(n − 2c + 2)! r
= ɛD 2c−2,c−1 + ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1
2 n!
r
= ɛ c−(n−r) D 2n−2r,n−r + (c − (n − r)) ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1
(A.15)
2 n!
r
= ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1
+ (c − (n − r)) ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1
2 n!
r
2 n!
r
= ɛc−(n−r) vol(G C n−1,r )
( ) n − 1 −1
(c + r − n + 1) .
2 n!
r
Thus, we get the value of the unknowns (in the vector x) until position D 2⌊n/2⌋,⌊n/2⌋ .
Second stage: 2c ≥ n. Note that B 2c,q ′ is defined if q ≥ 2c − n > 0. In this case, 2c ≥ n,
and the system we have to solve has the same structure as in the previous case (2c ≤ n) but
with less equations and unknowns. Taking the same equations as in the previous case 2c ≤ n,
we obtain, as augmented matrix,
⎛
(2c − n) ɛ((4c − 2n − 1)C 2c−2,2c−n−1−
−4(n − c + 1)(2n − 2c + 1)C 2c−2,2c−n−2)
⎜ (2n − 2c)
⎝
2 −(2c − n − 1)
⎟
⎠ .
4 −2c(n − c) −2ɛc(c − n)D 2c−2,c−1
⎞
r
) −1

98 Appendix
From the first equation, it follows
ɛ
C 2c,2c−n =
2c − n ((4c−2n−1)C 2c−2,2c−n−1 −2(2n−2c+2)(2n−2c+1)C 2c−2,2c−n−2 ). (A.16)
For a ∈ {0, ..., n − c}, by relation
we get
and from
C 2c,2c−n+a =
and (A.17) we get
(2n − 2c − 2a + 2) 2 C 2c,2c−n+a−1 = a(2c − n + a)C 2c,2c−n+a
=
=
4(n − c − a + 1)2
C 2c,2c−n+a−1
a(2c − n + a)
4 a (n − c − a + 1) 2 (n − c − a + 2) 2 . . . (n − c) 2
a(a − 1) . . . 2(2c − n + a)(2c − n + a − 1) . . . (2c − n + 1) C 2c,2c−n
4 a (n − c)!(n − c)!(2c − n)!
a!(n − c − a)!(n − c − a)!(2c − n + a)! C 2c,2c−n
4C 2c,c−1 − 2(n − c)cD 2c,c = −2ɛ(n − c)(c)D 2c−2,c−1
D 2c,c = ɛD 2c−2,c−1 +
2
c(n − c) C 2c,c−1
= ɛD 2c−2,c−1 + 2 · 4n−c−1 (n − c)!(n − c)!(2c − n)!
C 2c,2c−n
c(n − c)(n − c − 1)!(c − 1)!
= ɛD 2c−2,c−1 + 2 4n−c−1 (n − c)!(2c − n)!
C 2c,2c−n .
c!
(A.17)
(A.18)
In order to obtain the value of C 2c,2c−n we use the value of C 2c−2,2c−n−1 and C 2c−2,2c−n−2
if c ∈ {n − r, ..., ⌊n/2⌋}. We consider c 0 = ⌊ n+2
2 ⌋.
From the previous case 2c ≤ n we know the value of the unknowns C 2c0 −2,2c 0 −n−1,
C 2c0 −2,2c 0 −n−2 and D 2c0 −2,c 0 −1. For n even we have (we omit the analogous computation
for n odd)
C 2c0 −2,2c 0 −n−1 = C n,1 = ɛn/2−(n−r) vol(G C n−1,r )(n/2)!(n/2)! ( ) n − 1 −1
4 n/2−1 (A.19)
n!((n − 2)/2)!((n − 2)/2)! r
= ɛr−n/2 vol(G C n−1,r ) ( ) n − 1 −1
2 n n 2 ,
n!
r
C 2c0 −2,2c 0 −n−2 = C n,0 = ɛr−n/2 vol(G C n−1,r )
2 n n!
( ) n − 1 −1
,
r
D 2c0 −2,c 0 −1 = D n,n/2 = ɛn/2−(n−r) vol(G C n−1,r )
(r − n ( ) n − 1 −1
2 n!
2 + 1) .
r
Then
ɛ
C n+2,2 =
2c − n ((4c − 2n − 1)C 2c−2,2c−n−1 − 2(2n − 2c + 2)(2n − 2c + 1)C 2c−2,2c−n−2 )
= ɛr−n/2+1 vol(G C n−1,r )
( ) n − 1 −1
2 n+1 (3n 2 − 2n(n − 1))
n!
r
= ɛr−n/2+1 vol(G C n−1,r ) ( ) n − 1 −1
2 n+1 n(n + 2) .
n!
r

Proof of Theorem 4.3.5 99
Once we know the expression of C n+2,2 we find the value of C 2c,2c−n , for every c ∈ {⌊n/2⌋, . . . , n},
using the recurrence for C 2c,2c−n and C 2c−2,2c−n−1 . First, we have
(A.16)
C 2c,2c−n =
(A.17)
ɛ
2c − n ((4c − 2n − 1)C 2c−2,2c−n−1 − 2(2n − 2c + 2)(2n − 2c + 1)C 2c−2,2c−n−2 )
= ɛC 2c−2,2c−n−2
·
(
2c − n
(4c − 2n − 1)4(n − c + 1)!(n − c + 1)!(2c − n − 2)!
·
(n − c)!(n − c)!(2c − n − 1)!
(
4ɛ(n − c + 1) (4c − 2n − 1)(n − c + 1) − (2n − 2c + 1)(2c − n − 1)
=
2c − n
(2c − n − 1)
4ɛ(n − c + 1)c
=
(2c − n)(2c − n − 1) C 2c−2,2c−n−2.
)
− 4(n − c + 1)(2n − 2c + 1)
)
C 2c−2,2c−n−2
We go on with this recurrence until C ∗,∗−n with ∗ ≤ n+2
2
. In this case we know the value of
the constants, and we can find the value of C 2c,2c−n .
C 2c,2c−n = (4ɛ)c−(n+2)/2 c(c − 1) · ... · ((n + 4)/2)(n − c + 1)(n − c + 2) · ... · (n/2 − 1)
C n+2,2
(2c − n)(2c − n − 1) · ... · 4 · 3
Finally,
= (4ɛ)c−(n+2)/2 c!((n − 2)/2)!2
((n + 2)/2)!(n − c)!(2c − n)! C n+2,2
= (4ɛ)c−(n+2)/2 2 3 ( ) c ɛ r−n/2+1 vol(G C n−1,r ) ( ) n − 1 −1
(n + 2)n 2c − n 2 n+1 n(n + 2)
n!
r
= ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1 ( ) c
4 n−c . (A.20)
n! r 2c − n
and
4 a (n − c)!(n − c)!(2c − n)! ɛ c−(n−r) c!vol(G C n−1,r
C 2c,2c−n+a =
) ( n − 1
a!(n − c − a)!(n − c − a)!(2c − n + a)! 4 n−c (2c − n)!(n − c)!n! r
= ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1 ( )( )
n − c c
4 n−c−a n! r n − c − a 2c − n + a
) −1
= ɛD 2c−2,c−1 + 2 4n−c−1 (n − c)!(2c − n)! ɛ c−(n−r) c!vol(G C n−1,r ) ( n − 1
c! 4 n−c (2c − n)!(n − c)!n! r
= ɛD 2c−2,c−1 + ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1
2 n!
r
= ɛ c−n/2 D n,n/2 + (c − n/2) ɛc−(n−r) vol(G C n−1,r ) ( ) n − 1 −1
2 n!
r
= ɛc−(n−r) vol(G C n−1,r )
( ) n − 1 −1
(c + r − n + 1) .
2 n!
r
D 2c,c
(A.17) and (A.20)
) −1
To determine the value of d, the coefficient of δvol, we consider the last equation of the

100 Appendix
initial linear system
So,
d
(n − 1)! = −2ɛ2 (2n − 1)D 2n−4,n−2 − 4ɛC 2n−2,n−2 + 2ɛ(3n − 1)D 2n−2,n−1
= 2ɛ r (−(2n − 1)(r − 1) − (n − 1) + (3n − 1)r) vol(GC n−1,r ) ( n − 1
2 n! r
= ɛr vol(G C n−1,r ) ( ) n − 1 −1
n(r + 1) .
n!
r
( ) n − 1 −1
d = ɛ r vol(G C n−1,r)(r + 1) .
r
Now, we have to prove that the given solution satisfies all the equations we did not use to
solve the system. This is because we cannot ensure that the equation (A.10) has solution.
Let us study first the case 2c ≤ n. Consider the matrix on page 96. The rows we did not
use correspond to ˜B 2c−1,q ′ , q ∈ {1, ..., c − 1}. Suppose q ≠ c − 1. The equations given by this
row are ˜B 2c−1,q
′
−(2c − 2q)(2c − 2q − q)C 2c,q + (q + 1)(n − 2c + q + 3/2)C 2c,q+1
= −ɛ(2c − 2q)(2c − 2q − 1)C 2c−2,q−1 + ɛ(n − 2c + q + 2)(q + 1/2)C 2c−2,q .
For q = c − 1, the equation is
2ɛ 2 (n − c + 1)(c − 1/2)D 2c−4,c−2 + 2ɛC 2c−2,c−2 − 2ɛ((n − c + 1)(2c − 1/2) − c/2)D 2c−2,c−1
= −2C 2c,c−1 + 2c(n − 2 + 1/2)D 2c,c .
Substituting the value of each C ∗,· and D ∗,· given on page 97 we prove that the equations are
satisfied.
In the same way, we can prove that all equations appearing in the case 2c ≥ n are also
satisfied.
Finally, using again the relation in (4.12), we get the result with respect to {B k,q , Γ k,q }.
Proof of Theorem 4.4.1
The idea of the proof of this theorem is the same as for Theorem 4.3.5. In the same way, if the
variation δ X is the same in both sides, for all differentiable vector field X, then the expression
holds.
From the Gauss-Bonnet-Chern formula, we know that χ(Ω) can be written as the integral
over N(Ω) of a differential form O(2n)-invariant, and also U(n)-invariant. Thus, by Proposition
2.4.5 there exist constants C k,q , D k,q , d such that
) −1
χ(Ω) = ∑ k,q
⌊ n 2 ⌋
C k,q B k,q ′ (Ω) + ∑
D 2j,j Γ ′ 2j,j(Ω) + dvol(Ω)
j=1
(A.21)
where max{0, k − n} ≤ q < k/2 ≤ n, and B
k,q ′ and Γ′ k,q
are the valuations defined in (4.12).
Taking the variation in both sides of the previous equality we have
0 = ∑ (c k,q ˜B′ k,q (Ω) + d 2q,q ˜Γ′ 2q,q (Ω))
k,q
with c k,q and d k,q linear combination of C k,q and D 2q,q .

Proof of Theorem 4.4.1 101
Thus, we have to impose c k,q = d k,q = 0.
The variation of Γ ′ 0,0 in CKn (ɛ) is (cf. Corollary 4.1.9)
δΓ ′ 0,0(Ω) = 2ɛ(−(3n − 1) ˜B ′ 1,0(Ω) + (n − 1)˜Γ ′ 1,0(Ω) + 3ɛ(n − 1) ˜B ′ 3,1(Ω)).
It is necessary to cancel the variation of the terms ˜B 1,0 ′ , ˜B 3,1 ′ and ˜Γ ′ 1,0 . By Proposition 4.1.7 we
have that the variation of a valuation B
k,q ′ in CKn (ɛ) with k even (resp. odd) has only terms
˜B
k ′ ′ ,q
and ˜Γ ′ ′ k ′ ,q
with k ′ odd (resp. even). Thus, in the expresssion (A.21) we can restrict the
′
value of k to k even, and (A.21) can be reduced to
⎛
n−1
∑
χ(Ω) = ⎝
k=0
∑k−1
q=max{0,2k−n}
C 2k,q B ′ 2k,q (Ω) + D 2k,kΓ ′ 2k,k (Ω) ⎞
⎠ + dvol(Ω).
(A.22)
The right hand side in the previous equality coincides with the right hand side of (A.11) plus
the term D 0,0 Γ ′ 0,0 . Thus, the variation is very similar and the linear system we have to solve
will be also very similar to the one solved in Theorem 4.3.5. The only different equations are
the ones given by c 1,0 = 0, d 1,0 = 0 and c 3,1 = 0, that is
−ɛ(3n − 1)D 0,0 − 2C 2,0 + 2(n − 1/2)D 2,1 = 0,
ɛ(n − 1)D 0,0 + 2C 2,0 − (n − 1)D 2,1 = 0,
3ɛ 2 (n − 1)D 0,0 + 2ɛC 2,0 − ɛ(7n − 9)D 2,1 − 2C 4,1 + 2(2n − 3)D 4,2 = 0.
(A.23)
We find the value of D 0,0 for ɛ = 0, i.e. in C n , using the Gauss-Bonnet formula so that
D 0,0 =
1
O 2n−1 (n − 1)! = 1
2nω 2n (n − 1)! = n!
2 n!π n = 1
2π n .
The choice for the value of D 0,0 ensures that both sides in (A.21) coincide when Ω collapses
to a point.
From the first two equation and the value of D 0,0 we get
C 2,0 = ɛ 2 (n − 1)D 0,0 =
D 2,1 = 2ɛD 0,0 = ɛ
π n .
ɛ(n − 1)
4π n ,
In order to find the value of C 4,1 and D 4,2 we consider the equations given by {c 3,0 =
0, c 3,1 = 0, d 3,0 = 0, d 3,1 = 0}. The equation c 3,1 = 0 is the one given in (A.23), and the others
are
−ɛ(n − 2)C 2,0 − 24C 4,0 + (2n − 5)C 4,1 = 0,
16C 4,0 − (n − 3)C 4,1 = 0,
ɛ(n − 2)D 2,1 + C 4,1 − (n − 2)D 4,2 = 0.
(Note that they coincide with the ones in Theorem 4.3.5.) Solving the system given by these
3 equations and (A.23), we get that it is compatible with solution
C 4,0 =
ɛ2
32π n (n − 2)(n − 3), C 4,1 = ɛ2
(n − 2),
2πn D
4,2 = 3ɛ2
2π n .
To find the value of the unknowns C 2c,q , D 2c,c with c ≥ 3, we have to solve the same
equations as in the proof of Theorem 4.3.5. We can use the same relations if we first prove
that C 4,0 , C 4,1 and D 4,2 also satisfies (A.12). We have to check it because variables C 4,0 , C 4,1

102 Appendix
and D 4,2 here were obtained solving another linear system. But, it is straightforward verified.
So, we get the same relation among the unknowns.
Thus, from the equalities (A.13), (A.14), (A.15), (A.17) and (A.18), with r = n − 1, and
the computation of C n,1 , C n,0 , D n,⌊n/2⌋ in the same way as in (A.19) we have
χ(Ω) =
n−1
∑
c=0
c=0
⎛
ɛ c
⎝
π n
∑c−1
q=max{0,2c−n}
1
4 c−q ( c
q
)( n − c
c − q
)
B ′ 2c,q(Ω) + c + 1
Using the relation in (4.12) we get the stated expression
⎛
n−1
∑ ɛ c ∑c−1
( )( )
χ(Ω) = ⎝
1 c n − c
π n 4 c−q B ′
q c − q
2c,q(Ω) + c + 1
=
=
=
n−1
∑
c=0
+
n−1
∑
c=0
n−1
∑
c=0
ɛ c (
π n
q=max{0,2c−n}
∑c−1
q=max{0,2c−n}
2(c + 1)
2
ɛ c (
π n
2 Γ′ 2c,c(Ω)
2 Γ′ 2c,c(Ω)
⎞
⎠+ ɛn (n + 1)!
vol(Ω).
⎞
π n
⎠ + ɛn (n + 1)!
vol(Ω)
1 c!(n − c)!q!(n − 2c + q)!(2c − 2q)!ω 2n−2c
4 c−q B 2c,q (Ω)+
q!(c − q)!(c − q)!(n − 2c + q)!
c!(n − 2c + c)!(2c − 2c)!ω 2n−2c Γ 2c,c (Ω) ) + ɛn (n + 1)!
vol(Ω)
∑c−1
q=max{0,2c−n}
( )
1 2c − 2q c!(n − c)!π
n−c
4 c−q B 2c,q (Ω)
c − q (n − c)!
π n−c
+ (c + 1)!(n − c)!
(n − c)! Γ 2c,c(Ω) ) + ɛn (n + 1)!
π n vol(Ω)
⎛
ɛ c
π c c! ∑c−1
( )
⎝
1 2c − 2q
4 c−q c − q
q=max{0,2c−n}
π n
π n
B 2c,q (Ω) + (c + 1)Γ 2c,c (Ω) ⎠ + ɛn (n + 1)!
vol(Ω).
⎞
π n

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Notation
( , ) : Hermitian product in C n+1 defined at (1.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
( , ) ɛ : Hermitian product in CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
〈 , 〉 ɛ : Hermitian metric of CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
α i : real part of a dual form of a J-moving frame on CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . 16
α ij : real part of a connection form of a J-moving frame on CK n (ɛ) . . . . . . . . . . . . . . . . . . 16
B k,q (Ω) : Hermitian intrinsic volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
β i : imaginary part of a dual form of a J-moving frame on CK n (ɛ) . . . . . . . . . . . . . . . . . . . 16
β ij : imaginary part of a connection form of a J-moving frame on CK n (ɛ) . . . . . . . . . . . . 16
C n : standard Hermitian space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
CH n : complex hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
CK n (ɛ) : complex space form with constant holomorphic curvature 4ɛ . . . . . . . . . . . . . . . . . 8
CP n : complex projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
cos ɛ (α) : generalized cosine function with angle α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
cot ɛ (α) : generalized cotangent function with angle α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
D :
distribution defined by the normal vector and the complex structure in a hypersurface
in a Kähler manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
dL r : density of the space of complex planes with complex dimension r . . . . . . . . . . . . . . . 20
E : bisector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Γ k,q (Ω) : Hermitian intrinsic volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
G C n,r : Grassmannian of complex r-planes in C n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
G n,k,p (C n ) : Grassmannian of (k, p)-planes in C n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
H n : real hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
H : subspace of C n+1 which defines the points in CK n (ɛ) in the projective model . . . . . . 9
J : complex structure of a complex manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
K(V ) : convex compact domains in the vector space V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
107

108 Notation
L R r : space of totally real planes with dimension r in CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . 19
L R r : totally real plane in L R r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
L C r : space of complex planes with complex dimension r in CK n (ɛ) . . . . . . . . . . . . . . . . . . . 19
L C r : complex plane in L C r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
L (r)
q : space of complex q-planes contained in a fixed complex r-plane . . . . . . . . . . . . . . . . . 25
: space of complex r-planes containing a fixed complex q-plane . . . . . . . . . . . . . . . . . . 25
L C r[q]
M n×n (C) : square matrices of order n with complex valued entries . . . . . . . . . . . . . . . . . . . 11
M i (S) : i-th mean curvature integral of a hypersurface S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
M D i
(S) : i-th mean curvature integral restricted to the distribution D . . . . . . . . . . . . . . . . 40
µ k,q : Hermitian intrinsic volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
N(Ω) : normal fiber bundle of a domain Ω ⊂ CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
O n : area of the sphere of radius 1 in the standard Euclidean space of dimension n . . . 26
ϕ i : dual form of a J-moving frame on CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
ϕ ij : connection form of a J-moving frame on CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
P U(n) : projective unitary group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
RH n : real projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
S(CK n (ɛ)) : unit tangent bundle of CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
σ i (A) : i-th symmetric elementary function of the bilineal form A . . . . . . . . . . . . . . . . . . . . 30
sin ɛ (α) : generalized sine function with angle α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
U k,p : valuation defined on C n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
U(p, q) : unitary group of index p, q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
U ɛ (n) : isometry group of CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
V i (Ω) : i-th intrinsic volume of the domain Ω ⊂ R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Val(V ) : continuous translation invariant valuations on the vector space V . . . . . . . . . . . 31
Val k (V ) : homogeneous valuations of degree k in Val(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Val + (V ) : even valuations in Val(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Val − (V ) : odd valuations in Val(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Val G (V ) : invariant valuations under the action of the group G in Val(V ) . . . . . . . . . . . . 31
Val U(n) (C n ) : continuous translation and U(n)-invariant valuations on C n . . . . . . . . . . . . 34
ω n : volume of the ball of radius 1 in the standard Euclidean space of dimension n . . . 38
ω i0 : dual form of an orthonormal moving frame on CK n (ɛ) . . . . . . . . . . . . . . . . . . . . . . . . . . 16
ω ij : connection form of an orthonormal moving frame on CK n (ɛ) . . . . . . . . . . . . . . . . . . . . 16

Index
2-point homogeneous space, 14
3-point homogeneous space, 14
affine surface area, 30
Alesker, 29, 32, 34, 36, 37, 58
almost complex structure, 7
angle between two planes, 9
Bergmann metric, 10
Bernig, 36–38, 61
bisector, 18, 82, 83
Blaschke, 29
cohomogenity 1 hypersurface, 83
coisotropic plane, 88
complex
hyperbolic space, 8
manifold, 7
projective space, 7, 8
space form, 7, 8, 14, 18, 36
spine of a bisector, 82
structure, 7
submanifold, 17
subspace, 17, 81
continuous valuation, 30
Crofton formula, 32
even valuation, 31
Fu, 36–38, 61
Fubini-Study metric, 10
generic position, 48
geodesic ball, 18
Hadwiger, 32
Hadwiger Theorem, 32, 36, 37
Hausdorff
distance, 30
metric, 30
Hermitian
intrinsic volume, 40, 61
product, 7
holomorphic
angle, 9
atlas, 7
curvature, 8
section, 8
homogeneous
hypersurface, 83
space, 13, 19
valuation of degree k, 31
horizontal lift, 10
intrinsic volume, 29–31
invariant
density, 19, 20
with respect to a group, 31
isometry group, 11, 14, 34
isotropy group of a complex r-plane, 19
J-bases, 14
J-moving frame, 14
(k, p)-plane, 81
Kähler
form, 8
manifold, 8
kinematic formula, 33
Lagrangian plane, 36, 90
linear
(k, p)-plane, 81
submanifold, 81
McMullen, 32
Theorem, 31
mean curvature integrals, 30, 31, 33
monotone valuation, 31
moving frame, 14
odd valuation, 31
Park, 38–40, 75
Quermasintegrale, 26
109

110 Index
real
hyperbolic space, 8
projective space, 8
space form, 8, 18
real spine of a bisector, 82
regular domain, 16
reproductive
formula, 33
property, 45, 56
restricted mean curvature integral, 40
r-pla complex, 17
r-pla totalment real, 17
Rumin
derivative, 61
operator, 61
second fundamental form, 30
slide of a bisector, 82
smooth valuation, 31
on a manifold, 37
space constant
holomorphic curvature, 7, 8, 14, 18
sectional curvature, 8, 18
spinal surface, 82
standard Hermitian space, 8
Steiner formula, 29, 30
symmetric elementary function, 30, 40
symplectic form, 38
template method, 33
totally
geodesic
submanifold, 17
subspace, 81
real
plane, 9
submanifold, 17
subspace, 17, 81
umbilical hypersurface, 18
translation invariant valuation, 31
trigonometric generalized functions, 10
unimodular group, 13
unit normal bundle, 16, 17
unit tangent bundle, 15
U(p, q), 11
valuation, 29